DEFINITION, EXISTENCE, STABILITY AND
UNIQUENESS OF THE SOLUTION TO A
SEMILINEAR ELLIPTIC PROBLEM WITH A
STRONG SINGULARITY AT u = 0
Daniela Giachetti, Martınez-Aparicio Pedro J., François Murat
To cite this version:
Daniela Giachetti, Martınez-Aparicio Pedro J., François Murat. DEFINITION, EXISTENCE,
STABILITY AND UNIQUENESS OF THE SOLUTION TO A SEMILINEAR ELLIPTIC
PROBLEM WITH A STRONG SINGULARITY AT u = 0. 2010 Mathematics Subject Classification: 35J25, 35J67, 35J75. 2016. <hal-01348682>
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Version 23 06 2016
DEFINITION, EXISTENCE, STABILITY
AND UNIQUENESS OF THE SOLUTION
TO A SEMILINEAR ELLIPTIC PROBLEM
WITH A STRONG SINGULARITY AT u = 0
DANIELA GIACHETTI, PEDRO J. MARTÍNEZ-APARICIO,
AND FRANÇOIS MURAT
Abstract. In this paper we consider a semilinear elliptic equation
with a strong singularity at u = 0, namely


in Ω,
u ≥ 0
−div A(x)Du = F (x, u) in Ω,


u=0
on ∂Ω,
with F (x, s) a Carathéodory function such that
0 ≤ F (x, s) ≤
h(x)
a.e. x ∈ Ω, ∀s > 0,
Γ(s)
with h in some Lr (Ω) and Γ a C 1 ([0, +∞[) function such that
Γ(0) = 0 and Γ0 (s) > 0 for every s > 0.
We introduce a notion of solution to this problem in the spirit
of the solutions defined by transposition. This definition allows us
to prove the existence and the stability of this solution, as well as
its uniqueness when F (x, s) is nonincreasing in s.
Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3 Definition of a solution to problem (2.1) . . . . . . . . . . . . . . . . . . . . . . . . 14
3.1 The space V(Ω) of test functions . . . . . . . . . . . . . . . . . . . . . . . . . . . .14
3.2 Definition of a solution to problem (2.1) . . . . . . . . . . . . . . . . . . . . 16
4 Statements of the existence, stability and uniqueness results . . . . 20
Key words and phrases. Semilinear equations, singularity at u = 0, existence,
stability, uniqueness.
2010 Mathematics Subject Classification. 35J25, 35J67, 35J75.
1
2
D. GIACHETTI, P.J. MARTÍNEZ-APARICIO, AND F. MURAT
5 A priori estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
5.1 A priori estimate of Gk (u) in H01 (Ω) . . . . . . . . . . . . . . . . . . . . . . . . 22
5.2 A priori estimate of ϕDTk (u) in (L2 (Ω))N for ϕ ∈ H01 (Ω)∩L∞ (Ω)
. . . . . . . . . . . . . . . . . . . . . . . . . . . .Z. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .25
F (x, u)v when δ is small . . . . . 31
5.3 Control of the quantity
{u≤δ}
5.4 A priori estimate of β(Tk (u)) in H01 (Ω) . . . . . . . . . . . . . . . . . . . . . 36
6 Proofs of the Stability Theorem 4.2 and of the Existence
Theorem 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
6.1 Proof of the Stability Theorem 4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . 39
6.2 Proof of the Existence Theorem 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . 46
7 Comparison Principle and proof of the Uniqueness Theorem 4.3 46
8 The case with a zeroth-order term µu with µ ∈ H −1 (Ω) ∩ M+
b (Ω) 50
8.1 Recalling the variational framework for the case with a
zeroth-order term µu with µ ∈ H −1 (Ω) ∩ M+
b (Ω) . . . . . . . . . . . . . . . 50
8.2 The adaptation of Definition 3.6 to the case with a
zeroth-order term µu with µ ∈ H −1 (Ω) ∩ M+
b (Ω) . . . . . . . . . . . . . . . 51
8.3 A counterexample to the strong maximum principle in the case
with a zeroth-order term µu with µ ∈ H −1 (Ω) ∩ M+
b (Ω) . . . . . . . . 54
Appendix A An useful lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Appendix B Three remarks about Definition 3.6 . . . . . . . . . . . . . . . . . . 57
B.1 Equivalence of two definitions in the case of a mild
singularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
B.2 Equivalence of two variants of Definition 3.6 using the
requirement “for every k” and the requirement “for a given k0 ” . 59
B.3 A variant of Definition 3.6 using another space W(Ω) of test
functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
Appendix C Study of the set where the solution u is equal
to zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
SEMILINEAR PROBLEMS WITH A STRONG SINGULARITY AT u = 0
3
1. Introduction
Position of the problem
In the present paper we deal with a semilinear problem with a strong
singularity at u = 0, which consists in finding a function u which
satisfies


in Ω,
u ≥ 0
(1.1)
−div A(x)Du = F (x, u) in Ω,

u = 0
on ∂Ω,
where Ω is a bounded open set of RN , N ≥ 1, where A is a coercive
matrix with coefficients in L∞ (Ω), and where
F : (x, s) ∈ Ω × [0, +∞[→ F (x, s) ∈ [0, +∞]
is a Carathéodory function which satisfies
(1.2)
0 ≤ F (x, s) ≤
h(x)
a.e. x ∈ Ω, ∀s > 0,
Γ(s)
with
(1.3)

h ≥ 0,



h ∈ Lr (Ω), r = 2N if N ≥ 3, r > 1 if N = 2, r = 1 if N = 1,
N +2

Γ
:
s
∈
[0,
+∞[−→
Γ(s) ∈ [0, +∞[ is a C 1 ([0, +∞[) function



such that Γ(0) = 0 and Γ0 (s) > 0 ∀s > 0.
The function F
A model for the function F (x, s) (see another possible model in (2.2)
below) is given by

f (x)
1
g(x)
F (x, s) =
2 + sin( ) + γ + l(x)
1
s
s
exp(− exp( s ))
(1.4)

a.e. x ∈ Ω, ∀s > 0,
where γ > 0 and where the functions f , g and l are nonnegative and
1
1
belong to Lr (Ω). In this model
1 , as well as γ , are just
s
exp(− exp( s ))
1
examples of functions which can be replaced by any singularity
Γ(s)
with Γ satisfying (1.3). Note that the behaviour of F (x, s) for s = 0
can be very different according to the point x.
Note also that the function F (x, s) is defined only for s ≥ 0, and
that in view of (1.2) and (1.3), F (x, s) is finite for almost every x ∈ Ω
4
D. GIACHETTI, P.J. MARTÍNEZ-APARICIO, AND F. MURAT
and for every s > 0, but that it can exhibit a singularity when s > 0
tends to 0 (and when h(x) > 0). Let us emphasize the fact that the
Carathéodory character of the function F (x, s), which can take infinite
values when s = 0, means in particular that for almost every x ∈ Ω,
the function s → F (x, s) is continuous not only for every s > 0 but
also for s = 0.
Note finally that we do not require F (x, s) to be nonincreasing in
s, except when we deal with comparison and uniqueness results in the
Uniqueness Theorem 4.3 and in Section 7 below.
The case of a mild singularity
In the present paper we consider the case of strong singularities,
namely the case where F (x, s) can have any (wild) behavior as s tends
to zero, while in our previous paper [8] we restricted ourselves to the
case of mild singularities, namely the case where
(1.5)

0 ≤ F (x, s) ≤ h(x) 1 + 1 a.e. x ∈ Ω, ∀s > 0
sγ

with 0 < γ ≤ 1.
When the function F satisfies (1.5), our definition of the solution
to problem (1.1) is relatively classical, because (see [8, Section 3]) it
consists in looking for a function u such that
(1.6 mild )

u ∈ H01 (Ω),





u ≥ 0 a.e. in Ω,

Z
F (x, u)ϕ < +∞ ∀ϕ ∈ H01 (Ω), ϕ ≥ 0,



ZΩ
Z



 ADuDϕ =
F (x, u)ϕ ∀ϕ ∈ H01 (Ω).
Ω
Ω
Definition of the solution for a strong singularity
In contrast, the definition of the solution to problem (1.1) that we
use in the present paper is less usual. This definition is in our opinion
one of the main originalities of the present paper. We refer the reader
to Section 3 below where this definition is given in details, but we
emphasize here some of its main features.
SEMILINEAR PROBLEMS WITH A STRONG SINGULARITY AT u = 0
5
When the function F satisfies (1.2) and (1.3), we will say (see Definition 3.6 below) that u is a solution to problem (1.1) if

1
i) u ∈ L2 (Ω) ∩ Hloc
(Ω),



ii) u(x) ≥ 0 a.e. x ∈ Ω,
(1.7 strong )

iii) Gk (u) ∈ H01 (Ω) ∀k > 0,



iv) ϕTk (u) ∈ H01 (Ω) ∀k > 0, ∀ϕ ∈ H01 (Ω) ∩ L∞ (Ω),
where Gk (s) = (s − k)+ , and if
(1.8 strong )

∀v ∈ H01 (Ω) ∩ L∞ (Ω), v ≥ 0,


X


t

with
−
div
A(x)Dv
=
ϕ̂i (−div ĝi ) + fˆ in D0 (Ω),




i∈I


1
∞


where
I
is
finite,
ϕ̂
∈
H
ˆi ∈ (L2 (Ω))N , f̂i ∈ L1 (Ω),
i
0 (Ω) ∩ L (Ω), g




one has


 Z
i)
F (x, u)v < +∞,

Ω

Z
Z

XZ


t


A(x)DvDGk (u) +
gˆi D(ϕ̂i Tk (u)) + fˆTk (u) =
ii)



Ω
Ω
i∈I Ω



t
t

= h−div
A(x)Dv, Gk (u)iH −1 (Ω),H01 (Ω) + hh−div A(x)Dv, Tk (u)iiΩ =


Z



=
F (x, u)v ∀k > 0,
Ω
where hh , iiΩ is a notation introduced in (3.3) below.
Note that in this definition every term of (1.8 strong ii) makes sense
in view of (1.7 strong iii) and (1.7 strong iv).
This definition (1.7 strong ), (1.8 strong ) strongly differs from the definition (1.6 mild ). Indeed for example the assertion u ∈ H01 (Ω) of (1.6 mild )
1
is replaced in (1.7 strong ) by u ∈ Hloc
(Ω), Gk (u) ∈ H01 (Ω) for every k > 0
1
and ϕTk (u) ∈ H0 (Ω) for every k > 0 and for every ϕ ∈ H01 (Ω)∩L∞ (Ω).
Here the assertion Gk (u) ∈ H01 (Ω) for every k > 0 in particular expresses the boundary condition u = 0 on ∂Ω (see Remark 3.7 ii))
below.
Actually, the solution u does not in general belong to H01 (Ω), or in
other terms u “does not belong to H 1 (Ω) up to the boundary”, when
the function F exhibits a strong singularity at s = 0. It is indeed
proved by A.C. Lazer and P.J. McKenna in [12, Theorem 2] that when
f ∈ C α (Ω), α > 0, with f (x) ≥ f0 > 0 in Ω, in a domain Ω with C 2+α
boundary, the solution u to the equation
6
D. GIACHETTI, P.J. MARTÍNEZ-APARICIO, AND F. MURAT


−∆u =
u = 0
f (x)
uγ
in Ω,
on ∂Ω,
does not belong to H 1 (Ω) when γ > 3, even if u belongs to
C 2+α (Ω) ∩ C 0 (Ω) in this case.
The “space” defined by (1.7 strong ) in which we look for a solution to
(1.1) in the case of a strong singularity is therefore fairly different from
the space H01 (Ω) in which we look for a solution to (1.1) in the case of
a mild singularity.
Another important difference between the two definitions appears
as far as the partial differential equation in (1.1) is concerned. Indeed, if the formulation in the last line of (1.6 mild ) is classical, with
the use of test functions in H01 (Ω), the equation in (1.1) as formulated in (1.8 strong ) involves test functions which belong to the vectorial
space described in the three first lines of (1.8 strong ), that we will denote by V(Ω) in the rest of the present paper. Actually this space
V(Ω) consists in functions v such that −div tA(x)Dv can be put in the
usual duality between H −1 (Ω) and H01 (Ω) with Gk (u), and in a “formal duality” (through the notation hh , iiΩ ) with Tk (u): indeed, when
u satisfies (1.7 strong ), one writes u as the sum u = Gk (u)+Tk (u), where
Gk (u) ∈ H01 (Ω) and where ϕTk (u) ∈ H01 (Ω) for every k > 0 and for
every ϕ ∈ H01 (Ω) ∩ L∞ (Ω); on the other hand −div tA(x)Dv, which of
course belongs to H −1 (Ω) when v ∈ H01 (Ω), will be assumed to be the
sum of a function fˆ ∈ L1 (Ω) and of a finite sum of functions ϕ̂i (−div gˆi ),
with ϕ̂i ∈ H01 (Ω) ∩ L∞ (Ω) and −div gˆi ∈ H −1 (Ω), a fact which allows
one to correctly define the “formal duality” hh−div tA(x)Dv, Tk (u)iiΩ
(see (3.3) in Definition 3.2 below).
This is a definition of the solution by transposition in the spirit of
those introduced by J.-L. Lions and E. Magenes and by G. Stampacchia. Here again things are fairly different with respect to the case of
a mild singularity.
It is worth noticing that nevertheless, as proved in Appendix B below, the two definitions that we use in the present paper (for a strong
singularity) and in [8] (for a mild singularity) coincide in the case of a
mild singularity.
Main results of the present paper
In the framework of the definition (1.7 strong ), (1.8 strong ) (i.e. of Definition 3.6 below) we are able to prove that there exists at least a
SEMILINEAR PROBLEMS WITH A STRONG SINGULARITY AT u = 0
7
solution to problem (1.1) (see the Existence Theorem 4.1 below). We
also prove that this solution is stable in the following sense: if un is a
solution to (1.1) for the function Fn (x, s), if the functions Fn (x, s) satisfy (1.2) for the same h and the same Γ, and if for some Carathédory
function F∞ (x, s) one has
(1.9) a.e. x ∈ Ω, Fn (x, sn ) → F∞ (x, s∞ ) if sn → s∞ , sn ≥ 0, s∞ ≥ 0,
then a subsequence of un tends in a convenient sense (actually in the
space defined by (1.7 strong )) to u∞ which is a solution to (1.1) for
the function F∞ (x, s) (see the Stability Theorem 4.2 below). Finally,
if further to (1.2) and (1.3) we assume that the function F (x, s) is
nonincreasing with respect to s, then this solution is unique (see the
Uniqueness Theorem 4.3 below).
In brief Definition 3.6 below provides a framework where problem
(1.1) is well posed in the sense of Hadamard.
Moreover every solution u to problem (1.1) defined in this sense
satisfies the following a priori estimates:
• an a priori estimate on Gk (u) in H01 (Ω) for every k > 0 (see Proposition 5.1 below) which is formally obtained by using Gk (u) as test
function; this implies an a priori estimate on u in L2 (Ω) (see Remark 5.2 below);
• an a priori estimate of ϕDTk (u) in (L2 (Ω))N for every
ϕ ∈ H01 (Ω) ∩ L∞ (Ω) (see Proposition 5.4 below) which is formally
obtained by using ϕ2 Tk (u) and ϕ2 as test functions; this implies an
1
a priori estimate on u in Hloc
(Ω) (see Remark 5.7 below) and an a
priori estimate on ϕTk (u) in H01 (Ω) for every k > 0 and for every
ϕ ∈ H01 (Ω) ∩ L∞ (Ω) (see Remark 5.5); Z
• an a priori estimate of the quantity
F (x, u)v for every
{u≤δ}
v ∈ V(Ω), v ≥ 0, and every δ > 0, which depends only on δ and
on v (and also on u through the products ϕ̂i Du) by a constant which
tends to zero when δ tends to zero (see Proposition 5.9 below);
• an a priori estimate of β(Tk (u)) in H01 (Ω) for every k >Z0 (where the
sp
function β is defined from the function Γ by β(s) =
Γ0 (t)dt)
0
(see Proposition 5.12 below) which is formally obtained by using
Γ(Tk (u)) as test function.
The (mathematically correct) proofs of all the above a priori estimates
are based on Definition (1.7 strong ), (1.8 strong ) and on the use of convenient test functions v in (1.8 strong ).
8
D. GIACHETTI, P.J. MARTÍNEZ-APARICIO, AND F. MURAT
The strategy of the present paper consists in following the method
used in our paper [8], but its implementation is rather different. In
our paper [10] we continue the present work by studying the homogenization of the problem in an open set perforated with many small
holes in the case of a strong singularity (we treated the case of a mild
singularity in [8, Section 5]).
It is worth noticing that the definition (1.7 strong ), (1.8 strong ), as well
as all the results that it allows us to obtain in the present paper (except Proposition C.3 and Remark C.4 in Appendix C below) continue
to hold true (with convenient adaptations) when a zeroth-order term
µu, with µ a nonnegative bounded Radon measure which also belongs
to H −1 (Ω), is added to the left-hand side of the partial differential
equation in (1.1), i.e. when problem (1.1) is replaced by

u≥0
in Ω,



−div A(x)Du + µu = F (x, u)
in Ω,
(1.10)

u=0
on ∂Ω,



with µ a bounded Radon measure, µ ≥ 0, µ ∈ H −1 (Ω).
In Section 8 below we present the variations which should be made in
this context. For the comfort of the reader, we have decided to treat this
case in the final section rather than writing the whole of the present
paper in this context, since the latest choice would make the paper
more difficult to read. Note that in this context the strong maximum
principle does not hold true in general (see Counterexample 8.3 below).
Description of the proofs of the present paper
As far as the proofs of the results described above are concerned, we
begin in the present paper by proving (see Section 5 below) that every
solution to problem (1.1) in the sense of Definition 3.6 satisfies the four
a priori estimates that we just mentioned.
We then prove (see Subsection 6.1 below) the Stability Theorem 4.2
by using these estimates, by extracting a subsequence and by passing
to the limit in (1.7 strong ) andZ (1.8 strong ). A keypoint in this proof is
the estimate of the quantity
F (x, un )v which implies that this
{un ≤δ}
term is uniformly small in n for δ > 0 sufficiently small. This allow us
to pass to the limit in n in the equation (1.8 strong ii).
Then the Existence Theorem 4.1 follows as a consequence of the
Stability Theorem 4.2 applied to the functions Fn defined by
Fn (x, s) = Tn (F (x, s+ )) a.e. x ∈ Ω, ∀s ≥ 0,
SEMILINEAR PROBLEMS WITH A STRONG SINGULARITY AT u = 0
9
where Tn is the usual truncation at height n (see Subsection 6.2 below).
Finally we prove (see Section 7 below) the Uniqueness Theorem 4.3
as a simple consequence of a Comparison Principle stated in Proposition 7.1 below.
In the last Section of the present paper (see Section 8 below), we
consider the case where a zeroth-order term µu, with µ a nonnegative
bounded Radon measure which belongs to H −1 (Ω), is added to the
left-hand side of the partial differential equation (1.1).
References
There is a wide literature dealing with problem (1.1). We will not
try to give here a complete list of references and we will concentrate on
a very few papers, also refering the interested reader to the references
quoted there.
Our work has been inspired by the paper [2] of L. Boccardo and
L. Orsina. In this paper, the authors address the problem (1.1) with
f (x)
F (x, s) = γ , where γ > 0 and where f ≥ 0 belongs to L1 (Ω) or
s
to other Lebesgue’s spaces, or to the space of Radon’s measures, and
they prove existence and regularity as well as non existence results. In
this work the strong maximum principle and the nonincreasing charf (x)
acter of the function F (x, s) = γ with respect to s play prominent
s
roles. The solution u to problem (1.1) is indeed required to satisfy
u(x) ≥ c(ω, u) > 0 on every open set ω with ω ⊂ Ω, and the framework in which the solution is searched for is the Sobolev’s space H01 (Ω)
1
(Ω).
or Hloc
In their paper [1], L. Boccardo and J. Casado-Dı́az continue along
the same lines and prove various properties of the solution, including
the uniqueness of the solutions obtained by approximation and the
stability of the solution with respect to the G-convergence a sequence
of matrices Aε (x) which are equicoercive and equibounded.
Less recent but important papers are those of C.A. Stuart [15], of
M.G. Crandall, P.H. Rabinowitz and L. Tartar [3] and of A.C. Lazer
and P.J. McKenna [12], which mainly work in C 2,α (Ω) or W 2,q (Ω)
frameworks and use methods of sub- and super-solutions. In particular
f (x)
A.C. Lazer and P.J. McKenna prove in [12] that when F (x, s) = γ
s
with γ > 0, f ∈ C α (Ω) and f (x) ≥ f0 > 0 in a C 2+α domain Ω, then
γ+1
one has c1 φ1 (x) ≤ u(x) 2 ≤ c2 φ1 (x) for two constants 0 < c1 < c2 ,
where φ1 is the first (positive) eigenfunction of −div A(x)D in H01 (Ω),
and M.G. Crandall, P.H. Rabinowitz and L. Tartar study in [3] the
10
D. GIACHETTI, P.J. MARTÍNEZ-APARICIO, AND F. MURAT
behaviours of u(x) and |Du(x)| at the boundary. Let us finally note
that C. Stuart in [14], as well as M.G. Crandall, P.H. Rabinowitz and
L. Tartar in [3], do not assume that F (x, s) is nonincreasing in s.
Strong and weak points of the present paper
Le us conclude this Introduction by emphasizing what are in our
opinion the strong and the weak points of the present paper.
One of the strong points is Definition 3.6. This definition makes
problem (1.1) well posed in the sense of Hadamard, and allows us to
perform in a mathematically correct way all the formal computations
that we want to make on problem (1.1). Two other strong points are the
fact that we do not assume that the function F (x, s) is nonincreasing
with respect to s, and that we do no not use the strong maximum
principle: indeed this property can be unavailable in situations which
are close to the problem that we consider in the present paper.
A weak point of the present paper is the fact that Definition 3.6 is
a definition by transposition in the spirit of J.-L. Lions and E. Magenes and of G. Stampacchia, a feature which makes difficult (if not
impossible) to extend it to the case of nonlinear monotone operators.
Let us finally announce that in [10] we treat in the present setting of
a strong singularity the homogenization of the problem (1.1) posed in
an open set Ωε obtained from Ω by removing many small holes, when
the Dirichlet boundary condition on the boundary ∂Ωε leads to the
appearance of a “strange term” µu in Ω. Here again Definition 3.6 plays
a crucial role, since this definition allows us to use the test functions
which are needed in the homogenization process.
2. Assumptions
We study in this paper solutions to the following singular semilinear
problem


in Ω,
u ≥ 0
(2.1)
−div A(x)Du = F (x, u) in Ω,

u = 0
on ∂Ω,
where a model for the function F (x, s) is given by (1.4); another possible model is the following

f (x)
1
g(x)
F (x, s) =
2 + sin( ) + γ + l(x)
1
s
s
exp(− exp(exp( s )))
(2.2)

a.e. x ∈ Ω, ∀s > 0,
where γ > 0 and where the functions f, g and l are nonnegative.
SEMILINEAR PROBLEMS WITH A STRONG SINGULARITY AT u = 0
11
In this section, we give the precise assumptions that we make on the
data of problem (2.1).
We assume that Ω is an open bounded set of RN , N ≥ 1 (no regularity is assumed on the boundary ∂Ω of Ω), that the matrix A is
bounded and coercive, i.e. satisfies
(2.3)
A(x) ∈ (L∞ (Ω))N ×N , ∃α > 0, A(x) ≥ αI a.e. x ∈ Ω,
and that the function F satisfies
(2.4)

F : (x, s) ∈ Ω × [0, +∞[→ F (x, s) ∈ [0, +∞] is a Carathéodory




function,
i.e. F satisfies



i) ∀s ∈ [0, +∞[, x ∈ Ω → F (x, s) ∈ [0, +∞] is measurable,



ii) for a.e. x ∈ Ω, s ∈ [0, +∞[→ F (x, s) ∈ [0, +∞] is continuous,
(2.5)

i) ∃h, h(x) ≥ 0 a.e. x ∈ Ω, h ∈ Lr (Ω),




2N


with r =
if N ≥ 3, r > 1 if N = 2, r = 1 if N = 1,



N +2
ii) ∃Γ : s ∈ [0, +∞[→ Γ(s) ∈ [0, +∞[, Γ ∈ C 1 ([0, +∞[),



such that Γ(0) = 0 and Γ0 (s) > 0 ∀s > 0,




h(x)

iii) 0 ≤ F (x, s) ≤
a.e. x ∈ Ω, ∀s > 0.
Γ(s)
Moreover, when we will prove comparison and uniqueness results
(Proposition 7.1 and Theorem 4.3), we will assume that F (x, s) is nonincreasing in s, i.e. that
(2.6)
F (x, s) ≤ F (x, t) a.e. x ∈ Ω, ∀s, ∀t, 0 ≤ t ≤ s.
Remark 2.1.
• i) If a function Γ = Γ(s) satisfies (2.5 ii), then Γ is (strictly) increasing and satisfies Γ(s) > 0 for every s > 0; note that the function Γ
can be either bounded or unbounded.
Observe also that if a function F = F (x, s) satisfies (2.5) for
h(x) and Γ(s), and if Γ(s) is a function which satisfies (2.5 ii) and
Γ(s) ≤ Γ(s), then F (x, s) satisfies (2.5) for h(x) and Γ(s).
• ii) The function F = F (x, s) is a nonnegative Carathéodory function
with values in [0, +∞] and not only in [0, +∞[. But, in view of
conditions (2.5 ii) and (2.5 iii), for almost every x ∈ Ω, the function
F (x, s) can take the value +∞ only when s = 0 (or, in other terms,
F (x, s) is finite for almost every x ∈ Ω when s > 0).
12
D. GIACHETTI, P.J. MARTÍNEZ-APARICIO, AND F. MURAT
• iii) Note that (2.5 ii) and (2.5 iii) do not impose any restriction on the
growth of F (x, s) as s tends to zero. Moreover it can be proved (see
[9, Section 3, Proposition 1]) that (2.5) is equivalent to the following
assumption


∀k > 0, ∃hk , hk (x) ≥ 0 a.e. x ∈ Ω,
(2.7)
hk ∈ Lr (Ω) with r as in (2.5 i) such that

0 ≤ F (x, s) ≤ h (x) a.e. x ∈ Ω, ∀s ≥ k.
k
• iv) Note that no growth condition is imposed from below on F (x, s) as
s tends to zero. Indeed it can be proved (see [9, Section 3, Remark 2])
that for every given functions G1 (s) and G2 (s) which satisfy (2.5),
G2 (s)
with h1 (x) = h2 (x) = 1, G1 (s) ≤ G2 (s) and
which tends
G1 (s)
to infinity as s tends to zero, there exist a function F (s) and two
sequences s1n and s2n which tend to zero such that F (s1n ) = G1 (s1n )
and F (s2n ) = G2 (s2n ).
Of course the growth of F (x, s) as s tends to zero can (strongly)
depend on the point x ∈ Ω.
• v) Note that the growth condition (2.5 iii) is stated for every s > 0,
while in (2.4) F is supposed to be a Carathéodory function defined
for s in [0, +∞[ and not only in ]0, +∞[. Indeed an indeterminacy
0
h(x)
appears in
when h(x) = 0 and s = 0, while the growth and
0
Γ(s)
Carathéodory assumptions imply that
F (x, s) = 0 ∀s ≥ 0 a.e. on {x ∈ Ω : h(x) = 0}.
In contrast, when h is assumed to satisfy h(x) > 0 for almost every
x ∈ Ω, one can write (2.5 iii) for every s ≥ 0.
• vi) Let us observe that the functions F (x, s) given in examples (1.4)
and (2.2) satisfy assumption (2.5); indeed for these examples one has
1
0 ≤ F (x, s) ≤ h(x)
+1
Γ(s)
for some h(x) and Γ(s) which satisfy (2.5 i) and (2.5 ii); taking
Γ(s)
Γ(s) =
it is clear that Γ(s) satisfies (2.5 ii) and that F (x, s)
1 + Γ(s)
satisfies (2.5) for h(x) and Γ(s).
• vii) The C 1 regularity of the function Γ which is assumed in (2.5 ii) is
used to define the function β which appears in Proposition 5.12 (see
(5.58)); this latest result of β(Tk (u)) is in turn strongly used in the
SEMILINEAR PROBLEMS WITH A STRONG SINGULARITY AT u = 0
13
proofs of the Comparison Principle of Proposition 7.1, of the Uniqueness Theorem 4.3, and of the equivalence results of Propositions B.1
and B.2 of Appendix B below.
This C 1 regularity could appear as a strong restriction on the
function Γ, but it can actually be proved (see [9, Section 3]) that,
given a function Γ such that for some δ > 0 and some M > 0
Γ ∈ C 0 ([0, +∞[), Γ(0) = 0, Γ(s) > 0 ∀s > 0, Γ(s) ≥ δ ∀s ≥ M,
one can construct a function Γ which satisfies (2.5 ii) as well as
Γ(s) ≤ Γ(s) for every s > 0 (see Remark 2.1 i) above).
Remark 2.2. The function h which appears in hypothesis (2.5 i) is
an element of H −1 (Ω). Indeed, when N ≥ 3, the exponent r = N2N
is
+2
∗ 0
∗
nothing but the Hölder’s conjugate (2 ) of the Sobolev’s exponent 2 ,
i.e.
1
1
1
1
1
(2.8)
when N ≥ 3,
= 1 − ∗ , where ∗ = − .
r
2
2
2 N
Making an abuse of notation, we will set
(
2∗ = any p with 1 < p < +∞ when N = 2,
(2.9)
2∗ = +∞ when N = 1.
∗ 0
With this abuse of notation, h belongs to Lr (Ω) = L(2 ) (Ω) ⊂ H −1 (Ω)
for all N ≥ 1 since Ω is bounded.
This result is indeed a consequence of Sobolev’s, Trudinger Moser’s
and Morrey’s inequalities, which (with this abuse of notation) assert
that
(2.10)
kvkL2∗ (Ω) ≤ CS kDvk(L2 (Ω))N
∀v ∈ H01 (Ω) when N ≥ 1,
where CS is a constant which depends only on N when N ≥ 3, but
which depends on p and on Q when N = 2, and which depends on Q
when N = 1, when Q is any bounded open set such that Ω ⊂ Q.
Remark 2.3. Assumption (2.6), namely the fact that for almost every
x ∈ Ω the function s → F (x, s) is nonincreasing in s, is not of the same
nature as assumptions (2.4) and (2.5) on F (x, s).
We will only use assumption (2.6) when proving comparison and
uniqueness results, namely Proposition 7.1 and Theorem 4.3. In contrast, all the others results of the present paper, and in particular the
existence and stability results stated in Theorems 4.1 and 4.2, and the
a priori estimates of Section 5, will not use this assumption.
14
D. GIACHETTI, P.J. MARTÍNEZ-APARICIO, AND F. MURAT
2.1. Notation
We denote by M+
b (Ω) the space of nonnegative bounded Radon measures on Ω.
We denote by D(Ω) the space of the functions C ∞ (Ω) whose support
is compact and included in Ω, and by D0 (Ω) the space of distributions
on Ω.
In the present paper, ϕ, ϕ and ψ will denote functions which belong
to H01 (Ω) ∩ L∞ (Ω), while φ and φ will denote functions which belong
to D(Ω).
Since Ω is bounded, kDwkL2 (Ω)N is a norm equivalent to kwkH 1 (Ω)
on H01 (Ω). We set
kwkH01 (Ω) = kDwk(L2 (Ω))N ∀w ∈ H01 (Ω).
For every s ∈ R and every k > 0 we define as usual
s+ = max{s, 0}, s− = max{0, −s},
Tk (s) = max{−k, min{s, k}}, Gk (s) = s − Tk (s).
For l : x ∈ Ω → l(x) ∈ [0, +∞] any measurable function we denote
{l = 0} = {x ∈ Ω : l(x) = 0}, {l > 0} = {x ∈ Ω : l(x) > 0}.
3. Definition of a solution to problem (2.1)
3.1. The space V(Ω) of test functions
In order to introduce the notion of solution to problem (2.1) that we
will use in the present paper, we define the following space V(Ω) of test
functions and a notation.
Definition 3.1. We define the space V(Ω) as the space of the functions
v which satisfy
(3.1)
(3.2)
v ∈ H01 (Ω) ∩ L∞ (Ω),


∃I finite, ∃ϕ̂i , ∃ĝi , i ∈ I, ∃fˆ, with



ϕ̂i ∈ H01 (Ω) ∩ L∞ (Ω), ĝi ∈ (L2 (Ω))N , fˆ ∈ L1 (Ω),
such that

X


t

−div
A(x)Dv
=
ϕ̂i (−div ĝi ) + fˆ in D0 (Ω).


i∈I
In the definition of V(Ω) we use the notation ϕ̂i , gˆi , and fˆ to help
the reader to identify the functions which enter in the definition of the
functions of V(Ω).
Note that V(Ω) is a vector space.
SEMILINEAR PROBLEMS WITH A STRONG SINGULARITY AT u = 0
15
Definition 3.2. When v ∈ V(Ω) with
X
−div tA(x)Dv =
ϕ̂i (−div ĝi ) + fˆ in D0 (Ω),
i∈I
where I, ϕ̂i , ĝi and fˆ are as in (3.2), and when z satisfies
1
z ∈ Hloc
(Ω) ∩ L∞ (Ω) with ϕz ∈ H01 (Ω), ∀ϕ ∈ H01 (Ω) ∩ L∞ (Ω),
we use the following notation:
t
hh−div A(x)Dv, ziiΩ =
(3.3)
XZ
Z
ĝi D(ϕ̂i z) +
Ω
i∈I
fˆz.
Ω
Remark
3.3. In (3.2),
the product ϕ̂i (−div ĝi ) with
1
∞
ϕ̂i ∈ H0 (Ω) ∩ L (Ω) and ĝi ∈ (L2 (Ω))N is, as usual, the distribution
on Ω defined, for every φ ∈ D(Ω), as
(3.4)

hϕ̂i (−div ĝi ), φiD0 (Ω),D(Ω)
= h−div
ĝi , ϕ̂i φiH −1 (Ω),H01 (Ω) =
Z

=
ĝi D(ϕ̂i φ),
Ω
and the equality −div tA(x)Dv =
X
ϕ̂i (−div ĝi ) + fˆ holds in D0 (Ω).
i∈I
In notation (3.3), the right-hand side is correctly defined since
ϕ̂i z ∈ H01 (Ω) and since z ∈ L∞ (Ω). In contrast the left-hand side
hh−div tA Dv, ziiΩ is just a notation.
Remark 3.4. If y ∈ H01 (Ω) ∩ L∞ (Ω), then ϕy ∈ H01 (Ω) for every
ϕ ∈ H01 (Ω) ∩ L∞ (Ω), so that for every v ∈ V(Ω), hh−div tA Dv, yiiΩ is
defined. Let us prove that actually one has
(
∀v ∈ V(Ω), ∀y ∈ H01 (Ω) ∩ L∞ (Ω),
(3.5)
hh−div tA(x)Dv, yiiΩ = h−div tA(x)Dv, yiH −1 (Ω),H01 (Ω) .
X
Indeed when −div tA(x)Dv =
ϕ̂i (−div ĝi ) + fˆ, one has for every
i∈I
φ ∈ D(Ω) (see (3.3) and (3.4))
Z

XZ
t


hh−div A(x)Dv, φiiΩ =
gˆi D(ϕ̂i φ) + fˆφ =



Ω

i∈I Ω
 X
ˆ
=h
ϕ̂i (−div gˆi ) + f , φiD0 (Ω),D(Ω) =


i∈I
Z



t

= h−div A(x)Dv, φiD0 (Ω),D(Ω) = tA(x)DvDφ,
Ω
16
D. GIACHETTI, P.J. MARTÍNEZ-APARICIO, AND F. MURAT
and therefore
Z
Z
XZ
ˆ
gˆi D(ϕ̂i φ) + f φ =
(3.6)
i∈I
Ω
Ω
t
A DvDφ,
∀φ ∈ D(Ω).
Ω
For every given function y ∈ H01 (Ω) ∩ L∞ (Ω), taking a sequence
φn ∈ D(Ω) such that
φn → y
in H01 (Ω) strongly, φn * y in L∞ (Ω) weakly-star,
and passing to the limit in (3.6) with φ = φn , we obtain (3.5).
Remark 3.5. Let us give some examples of functions which belong to
V(Ω).
• i) If ϕ1 , ϕ2 ∈ H01 (Ω) ∩ L∞ (Ω), then ϕ1 ϕ2 ∈ V(Ω). Indeed
ϕ1 ϕ2 ∈ H01 (Ω) ∩ L∞ (Ω), and in the sense of distributions, one has

−div tA(x)D(ϕ1 ϕ2 ) =




t
t

= −div(ϕ1 A(x)Dϕ2 ) − div(ϕ2 A(x)Dϕ1 ) =
(3.7)
= ϕ1 (−div tA(x)Dϕ2 ) − tA(x)Dϕ2 Dϕ1 +


+ϕ2 (−div tA(x)Dϕ1 ) − tA(x)Dϕ1 Dϕ2 =



= ϕ̂1 (−div ĝ1 ) + ϕ̂2 (−div ĝ2 ) + fˆ in D0 (Ω),
with ϕ̂1 = ϕ1 ∈ H01 (Ω) ∩ L∞ (Ω), ϕ̂2 = ϕ2 ∈ H01 (Ω) ∩ L∞ (Ω),
gˆ1 = tA(x)Dϕ2 ∈ L2 (Ω)N , gˆ2 = tA(x)Dϕ1 ∈ (L2 (Ω))N , and
fˆ = − tA(x)Dϕ2 Dϕ1 − tA(x)Dϕ1 Dϕ2 ∈ L1 (Ω).
• ii) In particular, if ϕ ∈ H01 (Ω) ∩ L∞ (Ω), then ϕ2 ∈ V(Ω), with
(3.8)
−div tA(x)Dϕ2 = ϕ̂(−div ĝ) + fˆ in D0 (Ω),
with ϕ̂ = 2ϕ ∈ H01 (Ω) ∩ L∞ (Ω), ĝ = tA(x)Dϕ ∈ (L2 (Ω))N and
fˆ = −2 tA(x)DϕDϕ ∈ L1 (Ω).
• iii) If ϕ ∈ H01 (Ω) ∩ L∞ (Ω) with supp ϕ ⊂ K, K compact, K ⊂ Ω,
then ϕ ∈ V(Ω). Indeed
(3.9)
−div tA(x)Dϕ = φ(−div tA(x)Dϕ) in D0 (Ω),
for every φ ∈ D(Ω), with φ = 1 on K.
• iv) In particular every φ ∈ D(Ω) belongs to V(Ω).
3.2. Definition of a solution to problem (2.1)
We now give the definition of a solution to problem (2.1) that we
will use in the present paper.
SEMILINEAR PROBLEMS WITH A STRONG SINGULARITY AT u = 0
17
Definition 3.6. Assume that the matrix A and the function F satisfy
(2.3), (2.4) and (2.5). We say that u is a solution to problem (2.1) if u
satisfies

1
i) u ∈ L2 (Ω) ∩ Hloc
(Ω),



ii) u(x) ≥ 0 a.e. x ∈ Ω,
(3.10)
iii) Gk (u) ∈ H01 (Ω) ∀k > 0,



iv) ϕTk (u) ∈ H01 (Ω) ∀k > 0, ∀ϕ ∈ H01 (Ω) ∩ L∞ (Ω),
(3.11)


∀v ∈ V(Ω), v ≥ 0,
X


t

with
−
div
A(x)Dv
=
ϕ̂i (−div ĝi ) + fˆ in D0 (Ω),




i∈I



∞
1

(Ω)
∩
L
(Ω), gˆi ∈ (L2 (Ω))N , fˆ ∈ L1 (Ω),
where
ϕ̂
∈
H

i
0




one has

 Z
F (x, u)v < +∞,
i)


Z
Ω
Z

XZ


t

ii)
A(x)DvDGk (u) +
gˆi D(ϕ̂i Tk (u)) + fˆTk (u) =



Ω
Ω

i∈I Ω



t
t

= h−div A(x)Dv, Gk (u)iH −1 (Ω),H01 (Ω) + hh−div A(x)Dv, Tk (u)iiΩ =

 Z



=
F (x, u)v ∀k > 0.
Ω
Remark 3.7.
• i) Definition 3.6 is a mathematically correct framework which gives
a meaning to the solution to problem (2.1); in contrast (2.1) is only
formal.
In this Definition 3.6, the requirement (3.10) is the ”space” (which
is not a vectorial space) to which the solution should belong, while
the requirement (3.11), and specially (3.11 ii), expresses the partial
differential equation in (2.1) in terms of (non standard) test functions
in the spirit of the solutions defined by transposition by J.-L. Lions
and E. Magenes and by G. Stampacchia.
• ii) Note that the statement (3.10 iii) formally contains the boundary
condition “u = 0 on ∂Ω”. Indeed Gk (u) ∈ H01 (Ω) for every k > 0
formally implies that “Gk (u) = 0 on ∂Ω”, i.e. “u ≤ k on ∂Ω” for
every k > 0, which implies “u = 0 on ∂Ω”.
• iii) Note also that in Section 5 below we will obtain a priori estimates
for every solution u to problem (2.1) in the sense of Definition 3.6 in
the various “spaces” which appear in (3.10):
18
D. GIACHETTI, P.J. MARTÍNEZ-APARICIO, AND F. MURAT
an a priori estimate of kukL2 (Ω) (see (5.3));
an a priori estimate of kukH 1 (Ω) (see (5.19));
loc
an a priori estimate of kGk (u)kH01 (Ω) for every k > 0 (see (5.1));
an a priori estimate of kϕTk (u)kH01 (Ω) for every k > 0 and every
ϕ ∈ H01 (Ω) ∩ L∞ (Ω) (see (5.17)).
• iv) Note finally that (very) formally, we have
Z

t

“h−div A(x)Dv, Gk (u)iH −1 (Ω),H01 (Ω) = (−div tA(x)Dv) Gk (u) =
Z
Ω

=
v (−divA(x)DGk (u))”,
–
–
–
–
Ω
Z

t

“hh−div A(x)Dv, Tk (u)iiΩ = (−div tA(x)Dv) Tk (u) =
Z
Ω

=
v (−divA(x)DTk (u)”,
Ω
so that (3.11 ii) formally means that
Z
Z
t
“ v (−div A(x)Du) =
F (x, u)v”,
Ω
Ω
i.e. that
“ − div A(x)Du = F (x, u)”,
since every v can be written as v = v + − v − with v + ≥ 0 and v − ≥ 0.
Observe that the above formal computation has no meaning in general, while (3.11 ii) has a perfectly correct mathematical sense when
v ∈ V(Ω) and when u satisfies (3.10).
The following Proposition 3.8 asserts that every solution to problem
(2.1) in the sense of Definition 3.6 is a solution to (2.1) in the sense of
distributions. Note that Proposition 3.8 does not say anything about
the boundary conditions satisfied by u (see also Remark 3.7 ii).
Proposition 3.8. Assume that the matrix A and the function F satisfy
(2.3), (2.4) and (2.5). Then for every solution to problem (2.1) in the
sense of Definition 3.6 one has
(3.12)
1
u ≥ 0 a.e. in Ω, u ∈ Hloc
(Ω), F (x, u) ∈ L1loc (Ω),
−div A(x)Du = F (x, u) in D0 (Ω).
(3.13)
Proof. Since every φ ∈ D(Ω) belongs to V(Ω) (see Remark 3.5 iv)),
assumption (3.11 i) implies that
Z
(3.14)
F (x, u)φ < +∞, ∀φ ∈ D(Ω), φ ≥ 0,
Ω
SEMILINEAR PROBLEMS WITH A STRONG SINGULARITY AT u = 0
19
which together with F (x, u) ≥ 0 implies that F (x, u) ∈ L1loc (Ω).
Therefore (3.12) holds true when u is a solution to problem (2.1) in
the sense of Definition 3.6.
For what concerns (3.13), for every φ ∈ D(Ω) and φ ∈ D(Ω) with
φ = 1 on supp φ, one has (see (3.9) with ϕ = φ)

t

h−div
A(x)Dφ, Gk (u)iH −1Z(Ω),H01 (Ω) + hh−div tA(x)Dφ, Tk (u)iiΩ =

Z



t
t
A(x)Dφ DGk (u) +
A(x)Dφ D(φTk (u)) =
=
ZΩ
Z
Ω



t

A(x)DφDu =
A(x)DuDφ,
=
Ω
Ω
1
where we have used the fact that u ∈ Hloc
(Ω) and that φ = 1 on
supp φ. This implies that when u is a solution to problem (2.1) in the
sense of Definition 3.6, one has
Z
Z
(3.15)
A(x)DuDφ =
F (x, u)φ ∀φ ∈ D(Ω), φ ≥ 0,
Ω
Ω
namely
−div A(x)Du ≥ F (x, u) in D0 (Ω).
But (3.15) also implies that
Z
Z
A(x)DuDφ =
F (x, u)φ, ∀φ ∈ D(Ω),
Ω
φ ≤ 0,
Ω
namely
−div A(x)Du ≤ F (x, u) in D0 (Ω).
This proves (3.13).
Remark 3.9. When u satisfies (3.10 i) and (3.10 iii), assertion
(3.10 iv) is equivalent to
ϕDu ∈ (L2 (Ω))N ∀ϕ ∈ H01 (Ω) ∩ L∞ (Ω).
(3.16)
1
Indeed, if z ∈ Hloc
(Ω) and ϕ ∈ H01 (Ω) ∩ L∞ (Ω), one has for every
k>0
(
ϕDz = ϕDTk (z) + ϕDGk (z) in (D0 (Ω))N ,
D(ϕTk (z)) = ϕDTk (z) + Tk (z)Dϕ in (D0 (Ω))N ,
and therefore
ϕDz − D(ϕTk (z)) = ϕDGk (z) − Tk (z)Dϕ in (D0 (Ω))N .
This in particular implies that
ϕDz − D(ϕTk (z)) belongs to (L2 (Ω))N
1
when z ∈ Hloc
(Ω), Gk (z) ∈ H01 (Ω) and ϕ ∈ H01 (Ω) ∩ L∞ (Ω).
20
D. GIACHETTI, P.J. MARTÍNEZ-APARICIO, AND F. MURAT
1
Therefore if z ∈ Hloc
(Ω), Gk (z) ∈ H01 (Ω) and ϕ ∈ H01 (Ω) ∩ L∞ (Ω),
one has
ϕDz ∈ (L2 (Ω))N ⇐⇒ D(ϕTk (z)) ∈ (L2 (Ω))N ,
which in view of Lemma A.1 below and of the inequality
−ϕk ≤ ϕTk (u) ≤ +ϕk
implies that
ϕDz ∈ (L2 (Ω))N ⇐⇒ ϕTk (z) ∈ H01 (Ω).
Taking z = u gives the equivalence between (3.10 iv) and (3.16). Remark 3.10. The reader will find in Appendix B three remarks concerning Definition 3.6: in Subsection B.1 (Proposition B.1), the equivalence of Definition 3.6 with the definition of a solution which we use
in our paper [8] in the case of a mild singularity; in Subsection B.2
(Proposition B.2), the equivalence of two variants of Definition 3.6
using respectively the requirement “for every k” and the requirement
“for a given k0 ”; in Subsection B.3, a variant of Definition 3.6 using
another space W(Ω) of test functions.
Remark 3.11. In view of (3.12), the function F (x, u(x)) is finite almost everywhere on the set {x ∈ Ω : u(x) > 0}. In Appendix C
below we therefore study the set {x ∈ Ω : u(x) = 0}. We prove in
this Appendix that actually this set is (up to a set of zero measure) a
subset of {x ∈ Ω : F (x, 0) < +∞} (see (C.3)), and even a subset of
{x ∈ Ω : F (x, 0) = 0} (see (C.6)). More than that, following a result
of L. Boccardo and L. Orsina [2], we observe that one has u(x) > 0
almost everywhere on Ω when F (x, 0) 6≡ 0 (see Remark C.4). Note
that the proof of the latest result uses the strong maximum principle,
which is not always available in frameworks close to the framework of
the present paper (see Counterexample 8.3 below).
4. Statements of the existence, stability
and uniqueness results
In this section we state results of existence, stability and uniqueness
of the solution to problem (2.1) in the sense of Definition 3.6.
Theorem 4.1. (Existence). Assume that the matrix A and the function F satisfy (2.3), (2.4) and (2.5). Then there exists at least one
solution u to problem (2.1) in the sense of Definition 3.6.
SEMILINEAR PROBLEMS WITH A STRONG SINGULARITY AT u = 0
21
The proof of Theorem 4.1 relies on a stability result (see Theorem 4.2
below) and on a priori estimates of kGk (u)kH01 (Ω) for every k > 0, of
Z
1
∞
kϕ DTk (u)k(L2 (Ω))N for every ϕ ∈ H0 (Ω)∩L (Ω) and of
F (x, u)v
{u≤δ}
for every δ > 0 and every v ∈ V(Ω), v ≥ 0. These a priori estimates
are satisfied by every solution u to problem (2.1) in the sense of Definition 3.6 (see Propositions 5.1, 5.4 and 5.9). This proof will be done
in Subsection 6.2.
Theorem 4.2. (Stability). Assume that the matrix A satisfies assumption (2.3). Let Fn be a sequence of functions and F∞ be a function
which all satisfy assumptions (2.4) and (2.5) for the same h and the
same Γ. Assume moreover that
(4.1) a.e. x ∈ Ω, Fn (x, sn ) → F∞ (x, s∞ ) if sn → s∞ , sn ≥ 0, s∞ ≥ 0.
Let un be any solution to problem (2.1)n in the sense of Definition 3.6,
where (2.1)n is the problem (2.1) with F (x, s) replaced by Fn (x, s).
Then there exists a subsequence, still labelled by n, and a function
u∞ , which is a solution to problem (2.1)∞ in the sense of Definition 3.6,
such that
(4.2)

1
un → u∞ in L2 (Ω) strongly, in Hloc
(Ω) strongly and a.e. in Ω,



G (u ) → G (u ) in H 1 (Ω) strongly ∀k > 0,
k n
k ∞
0

ϕT
(u
)
→
ϕT
(u
)
in
H01 (Ω) strongly ∀k > 0,
k n
k ∞



∀ϕ ∈ H01 (Ω) ∩ L∞ (Ω).
The proof of the Stability Theorem 4.2 will be done in Subsection 6.1.
Finally, the following uniqueness result is an immediate consequence
of the Comparison Principle stated and proved in Section 7 below. Note
that both the Uniqueness Theorem 4.3 and the Comparison Principle of
Proposition 7.1 are based on the nonincreasing character of the function
F (x, s) with respect to s.
Theorem 4.3. (Uniqueness). Assume that the matrix A and the
function F satisfy (2.3), (2.4) and (2.5). Assume moreover that the
function F (x, s) is nonincreasing with respect to s, i.e. satisfies assumption (2.6). Then the solution to problem (2.1) in the sense of
Definition 3.6 is unique.
Remark 4.4. When assumptions (2.3), (2.4), (2.5) as well as (2.6) hold
true, Theorems 4.1, 4.2 and 4.3 together assert that problem (2.1) is
well posed in the sense of Hadamard in the framework of Definition 3.6.
22
D. GIACHETTI, P.J. MARTÍNEZ-APARICIO, AND F. MURAT
5. A priori estimates
In this section we state and prove a priori estimates which are satisfied by every solution to problem (2.1) in the sense of Definition 3.6.
5.1. A priori estimate of Gk (u) in H01 (Ω)
Proposition 5.1. (A priori estimate of Gk (u) in H01 (Ω)). Assume that the matrix A and the function F satisfy (2.3), (2.4) and
(2.5). Then for every u solution to problem (2.1) in the sense of Definition 3.6, one has
(5.1)
kGk (u)kH01 (Ω) = kDGk (u)k(L2 (Ω))N ≤
CS khkLr (Ω)
∀k > 0,
α Γ(k)
where CS is the (generalized) Sobolev’s constant defined in (2.10).
Remark 5.2. Since Ω is bounded, using Poincaré’s inequality
kykL2 (Ω) ≤ CP (Ω)kDyk(L2 (Ω))N
(5.2)
∀y ∈ H01 (Ω),
and writing u = Tk (u)+Gk (u), one easily deduces from (5.1) that every
solution u to problem (2.1) in the sense of Definition 3.6 satisfies the
following a priori estimate in L2 (Ω)
(5.3)
1
kukL2 (Ω) ≤ k|Ω| 2 + CP (Ω)
CS khkLr (Ω)
∀k > 0.
α Γ(k)
Remark 5.3. (Formal proof of Proposition 5.1). Taking formally
Gk (u) as test function in (2.1) one obtains
Z
Z
A(x)DGk (u)DGk (u) =
F (x, u)Gk (u) ∀k > 0.
(5.4)
Ω
Ω
Estimate (5.1) then follows from the growth condition (2.5 iii) and
from the facts that Gk (u) = 0 on the set {u ≤ k}, and that Γ(s) is
nondecreasing, so that
0 ≤ F (x, u)Gk (u) ≤
h(x)
h(x)
Gk (u) ≤
Gk (u) a.e. x ∈ Ω,
Γ(u)
Γ(k)
and finally from the (generalized) Sobolev’s inequality (2.10).
This formal computation will be made mathematically correct in the
proof below.
SEMILINEAR PROBLEMS WITH A STRONG SINGULARITY AT u = 0
23
Proof of Proposition 5.1.
Define for every k and n with 0 < k < n the function Sk,n as


if 0 ≤ s ≤ k,
0
(5.5)
Sk,n (s) = s − k if k ≤ s ≤ n,

n − k if n ≤ s.
First step. In this step we will prove that
Sk,n (u) ∈ V(Ω).
(5.6)
Observe that for every n > k we have
0 ≤ Sk,n (u) ≤ Gk (u) and |DSk,n (u)| = χ{k≤u≤n} |Du| ≤ |DGk (u)|.
By (3.10 iii) and Lemma A.1 of Appendix A below this implies that
(5.7)
Sk,n (u) ∈ H01 (Ω) ∩ L∞ (Ω).
Let us prove that (5.6) holds true.
Indeed let ψk : R+ → R be a C 1 nondecreasing function such that
ψk (s) = 0 for 0 ≤ s ≤
k
and ψk (s) = 1 for s ≥ k.
2
1
Since u ∈ Hloc
(Ω) one has
Dψk (u) = ψk0 (u)Du = ψk0 (u)DG k (u) in D0 (Ω),
2
from which using (3.10 iii) we deduce that Dψk (u) ∈ (L2 (Ω))N . Therefore ψk (u) ∈ H 1 (Ω) ∩ L∞ (Ω), and then, using again (3.10 iii), the
inequality
4
0 ≤ ψk (u) ≤ G k (u),
k 4
4
(which results from k G k (s) ≥ 1 when s ≥ k2 ) and Lemma A.1 below,
4
we obtain
(5.8)
ψk (u) ∈ H01 (Ω) ∩ L∞ (Ω).
On the other hand, in view of (5.7) we have
(5.9)
−div tA(x)DSk,n (u) ∈ H −1 (Ω),
and one easily proves that
(5.10) −div tA(x)DSk,n (u) = ψk (u)(−div tA(x)DSk,n (u)) in D0 (Ω),
which implies (5.6) with ϕ̂ = ψk (u), ĝ = tA(x) DSk,n (u) and fˆ = 0.
24
D. GIACHETTI, P.J. MARTÍNEZ-APARICIO, AND F. MURAT
Second step. Since Sk,n (u) ∈ V(Ω) by (5.6) and since Sk,n (u) ≥ 0, we
can use Sk,n (u) as test function in (3.11 ii). We get
Z
Z
t


A(x)DSk,n (u)DGk (u) + tA(x)DSk,n (u)D(ψk (u)Tk (u)) =



Ω
Ω

= h−div tA(x)DS (u), G (u)i −1
1
+
k,n
k
H
(Ω),H0 (Ω)
t

+hh−div
A(x)DSk,n (u), Tk (u)iiΩ =

Z




=
F (x, u)Sk,n (u),
Ω
where the second term of the left-hand side is zero since
D(ψk (u)Tk (u)) = ψk (u)DTk (u) + Tk (u)Dψk (u) in D0 (Ω),
and since DSk,n (u) = 0 on {u ≤ k}, while DTk (u) = 0 and Dψk (u) = 0
on {u ≥ k}. This gives
Z
Z
(5.11)
A(x)DGk (u)DSk,n (u) =
F (x, u)Sk,n (u).
Ω
Ω
Third step. Since Sk,n (s) = 0 for s ≤ k, one has, using the growth
condition (2.5 iii) and the (generalized) Sobolev’s inequality (2.10)
Z
Z
h(x)



F (x, u)Sk,n (u) ≤
χ
Sk,n (u) ≤


Γ(u) {u>k}

Ω
Ω

 Z h(x)
khkLr ((Ω)
≤
Sk,n (u) ≤
kSk,n (u)kL2∗ (Ω) ≤
(5.12)

Γ(k)
Ω Γ(k)




khkLr ((Ω)


CS kDSk,n (u)k(L2 (Ω))N ∀k > 0, ∀n > k.
≤
Γ(k)
With the coercivity (2.3) of the matrix A, (5.11) and (5.12) imply
that
khkLr ((Ω)
(5.13)
αkDSk,n (u)k(L2 (Ω))N ≤ CS
∀k > 0, ∀n > k.
Γ(k)
Therefore Sk,n (u) is bounded in H01 (Ω) for k > 0 fixed independently
of n > k, and the left-hand side (and therefore the right-hand side) of
(5.11) is bounded independently of n > k. Since
(5.14)
Sk,n (u) * Gk (u) in H01 (Ω) weakly and a.e. x ∈ Ω as n → +∞,
applying Fatou’s Lemma to the right-hand side of (5.11) one deduces
that
Z
(5.15)
F (x, u)Gk (u) < +∞ ∀k > 0.
Ω
SEMILINEAR PROBLEMS WITH A STRONG SINGULARITY AT u = 0
25
Then using (5.14) in the left-hand side and Lebesgue’s dominated
convergence in the right-hand side of (5.11), one obtains (5.4).
Estimate (5.1) follows either from (5.13) and (5.14) or from (5.4). 5.2.
A priori estimate of ϕDTk (u) in (L2 (Ω))N
1
ϕ ∈ H0 (Ω) ∩ L∞ (Ω)
for
Proposition 5.4. Assume that the matrix A and the function F satisfy
(2.3), (2.4) and (2.5). Then for every u solution to problem (2.1) in
the sense of Definition 3.6 one has
(5.16)

kϕDTk (u)k2(L2 (Ω))N ≤



2
32k 2
CS2 khkLr (Ω)
2
2
≤ 2 kAk(L∞ (Ω))N ×N kDϕk(L2 (Ω))N + 2
kϕk2L∞ (Ω)
2

α
α
Γ(k)


∀k > 0, ∀ϕ ∈ H 1 (Ω) ∩ L∞ (Ω),
0
where the (generalized) Sobolev’s constant CS is defined in (2.10).
Remark 5.5. From the a priori estimate (5.16) and from
(
kD(ϕTk (u))k2(L2 (Ω))N = kϕDTk (u) + Tk (u)Dϕk2(L2 (Ω))N ≤
≤ 2kϕDTk (u)k2(L2 (Ω))N + 2kTk (u)Dϕk2(L2 (Ω))N ,
one deduces that every solution u to problem (2.1) in the sense of Definition 3.6 satisfies the following a priori estimate of ϕTk (u) in H01 (Ω)


kϕT (u)k2
= kD(ϕTk (u))k2(L2 (Ω))N ≤

 k 2 H01 (Ω)


64k

2
2

kAk(L∞ (Ω))N ×N + 2k kDϕk2(L2 (Ω))N +
≤
α2
(5.17)
2

CS2 khkLr (Ω)


kϕk2L∞ (Ω)
+ 2

2

α Γ(k)



∀k > 0, ∀ϕ ∈ H01 (Ω) ∩ L∞ (Ω).
Remark 5.6. Using estimates (5.1) on DGk (u) and (5.16) on ϕDTk (u)
in (L2 (Ω))N , as well as
ϕDu = ϕDTk (u) + ϕDGk (u),
and
|DTk (u)||DGk (u)| = 0 a.e. in Ω,
26
D. GIACHETTI, P.J. MARTÍNEZ-APARICIO, AND F. MURAT
one deduces that every solution u to problem (2.1) in the sense of Definition 3.6 satisfies the following a priori estimate of ϕDu in (L2 (Ω))N
(5.18)

kϕDuk2(L2 (Ω))N = kϕDTk (u)k2(L2 (Ω))N + kϕDGk (u)k2(L2 (Ω))N ≤



2
32k 2
CS2 khkLr (Ω)
2
2
≤ 2 kAk(L∞ (Ω))N ×N kDϕk(L2 (Ω))N + 2 2
kϕk2L∞ (Ω)
2

α
α
Γ(k)


∀k > 0, ∀ϕ ∈ H 1 (Ω) ∩ L∞ (Ω),
0
which, taking k = k0 for some k0 fixed or minimizing the right-hand
side in k, provides an a priori estimate of kϕDuk2(L2 (Ω))N which does not
depend on k; unfortunately minimizing in k does not give an explicit
constant for a general function Γ.
Remark 5.7. Since for every φ ∈ D(Ω) one has
D(φu) = φDu + (Tk (u) + Gk (u))Dφ,
which implies that
|D(φu)| ≤ |φDu| + k|Dφ| + kDφk(L∞ (Ω))N |Gk (u)|,
using the inequality (a + b + c)2 ≤ 3(a2 + b2 + c2 ), and the a priori
estimates (5.18) and (5.1) together with Poincaré’s inequality (5.2),
one deduces that every solution u to problem (2.1) in the sense of
1
Definition 3.6 satisfies the following a priori estimate of u in Hloc
(Ω)
(5.19)


kφuk2H 1 (Ω) = kD(φu)k2(L2 (Ω))N ≤


0


2
2

32k
CS2 khkLr (Ω)

2
2


≤
3
kAk
kDϕk
+
2
kϕk2L∞ (Ω) +
(L∞ (Ω))N ×N
(L2 (Ω))N

α2
α2 Γ(k)2
!
2 khk2 r

C
 2
L (Ω)


+k kDφk2L2 (Ω) + CP2 (Ω) S2
kDφk2(L∞ (Ω))N

2

α
Γ(k)



∀k > 0, ∀φ ∈ D(Ω),
which, taking k = k0 for some k0 fixed or minimizing the right-hand
side in k, provides an a priori estimate of kφuk2H 1 (Ω) for every fixed
0
φ ∈ D(Ω), i.e. an a priori estimate of kuk2H 1 (Ω) , which does not
loc
depend on k.
Remark 5.8. (Formal proof of Proposition 5.4). The computation that we will perform in the present Remark is formal. We will
make it mathematically correct in the proof below.
SEMILINEAR PROBLEMS WITH A STRONG SINGULARITY AT u = 0
27
The idea of the proof of Proposition 5.4 is to formally use ϕ2 Tk (u)
as test function in (2.1), where ϕ ∈ H01 (Ω) ∩ L∞ (Ω). We formally get
(5.20)
Z
Z
Z
A(x)DuDTk (u) ϕ2 + 2
Ω
F (x, u)ϕ2 Tk (u),
A(x)DuDϕ ϕTk (u) =
Ω
Ω
in the second term of which we write Du = DTk (u) + DGk (u). Using
the coercivity (2.3) of the matrix A we have
(5.21)

Z


α ϕ2 |DTk (u)|2 ≤




Z
 ΩZ
≤ 2 A(x)DTk (u)Dϕ ϕTk (u) + 2 A(x)DGk (u)Dϕ ϕTk (u) +


Ω
Z Ω



2

+ F (x, u)ϕ Tk (u).
Ω
In this inequality we will use the estimates
 Z
2 A(x)DT (u)Dϕ ϕT (u) ≤
k
k
(5.22)
Ω

≤ 2kkAk(L∞ (Ω))N ×N kDϕk(L2 (Ω))N kϕDTk (u)k(L2 (Ω))N ,
and
(5.23)  Z
2 A(x)DG (u)Dϕ ϕT (u) ≤
k
k
Ω

≤ 2kkAk(L∞ (Ω))N ×N kDϕk(L2 (Ω))N kϕkL∞ (Ω) kDGk (u)k(L2 (Ω))N .
On the other hand, since 0 ≤ Tk (u) ≤ k, and using formally ϕ2 as
test function in (2.1), we have
(5.24)

Z
Z

2

0≤
F (x, u)ϕ Tk (u) ≤ k F (x, u)ϕ2 =



ZΩ
Z Ω



2


= k A(x)DuDϕ = 2k A(x)DuDϕ ϕ =
ZΩ
Ω
Z

= 2k A(x)DTk (u)Dϕ ϕ + 2k A(x)DGk (u)Dϕ ϕ ≤



Ω
Ω



≤ 2kkAk(L∞ (Ω))N ×N kDϕk(L2 (Ω))N kϕDTk (u)k(L2 (Ω))N +



+2kkAk ∞ N ×N kDϕk 2 N kϕk ∞ kDG (u)k 2 N .
k
L (Ω)
(L (Ω))
(L (Ω))
(L (Ω))
28
D. GIACHETTI, P.J. MARTÍNEZ-APARICIO, AND F. MURAT
Collecting together (5.21), (5.22), (5.23) and (5.24) we obtain
(5.25)
 Z


α ϕ2 |DTk (u)|2 ≤



 Ω
≤ 4kkAk(L∞ (Ω))N ×N kDϕk(L2 (Ω))N kϕDTk (u)k(L2 (Ω))N +


+4kkAk(L∞ (Ω))N ×N kDϕk(L2 (Ω))N kϕkL∞ (Ω) kDGk (u)k(L2 (Ω))N



∀k > 0, ∀ϕ ∈ H 1 (Ω) ∩ L∞ (Ω).
0
Using Young’s inequality in the first term of the right-hand side of
(5.25), and the estimate (5.1) on kDGk (u)k(L2 (Ω))N in the second term,
provides an estimate on kϕDTk (u)k2(L2 (Ω))N .
Proof of Proposition 5.4.
First step. By (3.10 iv) we have ϕTk (u) ∈ H01 (Ω) ∩ L∞ (Ω), and
therefore, by Remark 3.5 i), we have
ϕ2 Tk (u) = ϕ ϕTk (u) ∈ V(Ω),
(5.26)
with

t
2

−div A(x)D(ϕ Tk (u)) =
= ϕ(−div tA(x)D(ϕTk (u))) − tA(x)D(ϕTk (u))Dϕ +

+(ϕT (u))(−div tA(x)Dϕ) − tA(x)DϕD(ϕT (u)) in D0 (Ω).
k
k
Since ϕ2 Tk (u) ∈ V(Ω) with ϕ2 Tk (u) ≥ 0 we can use v = ϕ2 Tk (u) as
test function in (3.11 ii). Denoting by j the value of k which appears
in (3.11 ii), we get
(5.27)
Z

t

A(x)D(ϕ2 Tk (u))DGj (u) +



ΩZ
Z



t

+
A(x)D(ϕTk (u))D(ϕTj (u)) − tA(x)D(ϕTk (u))Dϕ Tj (u) +



Ω

Z Ω

 Z
t
+
A(x)Dϕ D(ϕTk (u)Tj (u)) − tA(x)DϕD(ϕTk (u)) Tj (u) =

Ω
Ω


t
2

=
h−div
A(x)D(ϕ
T
(u)),
G
(u)i

k
j
H −1 (Ω),H01 (Ω) +


t
2


+hh−div
A(x)D(ϕ Tk (u)), Tj (u)iiΩ =

Z




=
F (x, u)ϕ2 Tk (u) ∀j > 0,
Ω
where we observe that the fourth term of the left-hand side makes
sense due to the fact that ϕTk (u) ∈ H01 (Ω) ∩ L∞ (Ω) by (3.10 iv), which
implies that (ϕTk (u))Tj (u) ∈ H01 (Ω) ∩ L∞ (Ω) again by (3.10 iv).
1
Since u ∈ Hloc
(Ω), we can expand (in L1loc (Ω)) the integrands of
each of the 5 terms of the left-hand side of (5.27). One obtains 13
SEMILINEAR PROBLEMS WITH A STRONG SINGULARITY AT u = 0
29
terms whose integrands belong to L1 (Ω) since DGk (u), ϕDTk (u) and
ϕDTj (u) belong to (L2 (Ω))N . A simple but tedious computation leads
from (5.27) to
(5.28)
Z
Z
2


A(x)DTj (u)DTk (u)ϕ +
A(x)DGj (u)DTk (u) ϕ2 +



Z
Ω
 ΩZ
+2 A(x)DTj (u)Dϕ ϕTk (u) + 2 A(x)DGj (u)Dϕ ϕTk (u) =

ZΩ
Ω



2
1

=
F (x, u)ϕ Tk (u) ∀ϕ ∈ H0 (Ω) ∩ L∞ (Ω), ∀j > 0.
Ω
Taking j = k gives, since one has |DGk (u)||DTk (u)| = 0 almost everywhere in Ω,
(5.29)
Z


A(x)DTk (u)DTk (u)ϕ2 +



Z
 ΩZ
+2 A(x)DTk (u)Dϕ ϕTk (u) + 2 A(x)DGk (u)Dϕ ϕTk (u) =

ZΩ
Ω



2
1

=
F (x, u)ϕ Tk (u) ∀ϕ ∈ H0 (Ω) ∩ L∞ (Ω),
Ω
which is to be compared with (5.20). This result can be formally obtained by taking ϕ2 Tk (u) as test function in (2.1), but the above described proof makes this mathematically correct. From (5.29) and the
coercivity (2.3) of the matrix A we deduce that
(5.30)
 Z

α ϕ2 |DTk (u)|2 ≤




Z
 ΩZ
≤ 2 A(x)DTk (u)Dϕ ϕTk (u) + 2 A(x)DGk (u)Dϕ ϕTk (u) +

Ω

Z Ω



2
1
+ F (x, u)ϕ Tk (u) ∀ϕ ∈ H0 (Ω) ∩ L∞ (Ω),
Ω
(compare with (5.21).)
Second step. As far as the first and the second terms of the right-hand
side of (5.30) are concerned, we have, as in (5.22), (5.23)
 Z
2 A(x)DT (u)Dϕ ϕT (u) ≤
k
k
(5.31)
Ω

≤ 2kkAk(L∞ (Ω))N ×N kDϕk(L2 (Ω))N kϕDTk (u)k(L2 (Ω))N ,
30
D. GIACHETTI, P.J. MARTÍNEZ-APARICIO, AND F. MURAT
and
(5.32)  Z
2 A(x)DG (u)Dϕ ϕT (u) ≤
k
k
Ω

≤ 2kkAk(L∞ (Ω))N ×N kDϕk(L2 (Ω))N kϕkL∞ (Ω) kDGk (u)k(L2 (Ω))N .
On the other hand, since for every ϕ ∈ H01 (Ω)∩L∞ (Ω), ϕ2 belongs to
V(Ω) (see Remark 3.5 ii) and since ϕ2 ≥ 0, using ϕ2 as test function in
(3.11 ii) gives

t
2
t
2
h−div
Z A(x)Dϕ , Gk (u)iH −1 (Ω),H01 (Ω) + hh−div A(x)Dϕ , Tk (u)iiΩ =
=
F (x, u)ϕ2 ,
Ω
which gives
Z
 Z

2 A(x)DGk (u)Dϕ ϕ + 2 A(x)DTk (u)Dϕ ϕ =
ZΩ
Ω
(5.33)

2
1
=
F (x, u)ϕ ∀ϕ ∈ H0 (Ω) ∩ L∞ (Ω).
Ω
Therefore we have, for the last term of the right-hand side of (5.30)
(5.34)
Z
Z

2

0≤
F (x, u)ϕ Tk (u) ≤ k F (x, u)ϕ2 =



Ω Z

ZΩ


= 2k A(x)DTk (u)Dϕ ϕ + 2k A(x)DGk (u)Dϕ ϕ ≤

Ω
Ω



∞
N
×N
2
N
≤
2kkAk
kDϕk
kϕDT

k (u)k(L2 (Ω))N +
(L (Ω))
(L (Ω))


+2kkAk(L∞ (Ω))N ×N kDϕk(L2 (Ω))N kϕkL∞ (Ω) kDGk (u)k(L2 (Ω))N ,
which has to be compared with (5.24) of the formal proof above.
Third step. Collecting together (5.30), (5.31), (5.32) and (5.34) we
have proved that
(5.35)
 Z


α ϕ2 |DTk (u)|2 ≤



 Ω
≤ 4kkAk(L∞ (Ω))N ×N kDϕk(L2 (Ω))N kϕDTk (u)k(L2 (Ω))N +


+4kkAk(L∞ (Ω))N ×N kDϕk(L2 (Ω))N kϕkL∞ (Ω) kDGk (u)k(L2 (Ω))N



∀k > 0, ∀ϕ ∈ H 1 (Ω) ∩ L∞ (Ω),
0
which has to be compared with (5.25) above.
SEMILINEAR PROBLEMS WITH A STRONG SINGULARITY AT u = 0
31
α 2
1 2
X +
Y in the first term and
2
2α
estimate (5.1) on kDGk (u)k(L2 (Ω))N in the second term gives
Using Young’s inequality XY ≤
(5.36)
α

kϕDTk (u)k2(L2 (Ω)N ≤


2


 8k 2
≤
kAk2(L∞ (Ω))N ×N kDϕk2(L2 (Ω))N +
α



CS khkLr (Ω)


,
+4kkAk(L∞ (Ω))N ×N kDϕk(L2 (Ω))N kϕkL∞ (Ω)
α Γ(k)
1
1 2
1 2
in which we use again Young’s inequality XY ≤
X +
Y . We
α
2α
2α
finally obtain
(5.37)
α
2

 2 kϕDTk (u)k(L2 (Ω)N ≤
2
16k 2
CS2 khkLr (Ω)
2
2

kAk
kϕk2L∞ (Ω) ,
≤
kDϕk
+

(L∞ (Ω))N ×N
(L2 (Ω))N
α
2α Γ(k)2
i.e. (5.16). This complete the proof of Proposition 5.4.
Z
F (x, u)v when δ is small
5.3. Control of the quantity
{u≤δ}
In this subsection we prove an estimate (see (5.40) below) which is
a key point in the proofs of our results.
For δ > 0, we define the function Zδ : s ∈ [0, +∞[→ Zδ (s) ∈ [0, +∞[
by


if 0 ≤ s ≤ δ,
1
s
(5.38)
Zδ (s) = − δ + 2 if δ ≤ s ≤ 2δ,

0
if 2δ ≤ s.
Proposition 5.9. Assume that the matrix A and the function F satisfy
(2.3), (2.4) and (2.5). Then for every u solution to problem (2.1) in
the sense of Definition 3.6 and for every v such that

v ∈ V(Ω), v ≥ 0,


X

t
with
−
div
A(x)Dv
=
ϕ̂i (−div ĝi ) + fˆ in D0 (Ω)
(5.39)

i∈I


1
∞
where ϕ̂i ∈ H0 (Ω) ∩ L (Ω), gˆi ∈ L2 (Ω)N , fˆ ∈ L1 (Ω),
32
D. GIACHETTI, P.J. MARTÍNEZ-APARICIO, AND F. MURAT
one has
Z




∀δ > 0, {u≤δ} F (x, u)v ≤
!
Z X
XZ
3

ˆ

Zδ (u)gˆi Du ϕ̂i .
gˆi Dϕ̂i + f δ +

≤ 2
Ω i∈I
i∈I Ω
(5.40)
Remark 5.10. (Formal proof of Proposition 5.9). Estimate (5.40)
can be formally obtained by using in (2.1) the test function Zδ (u)v,
where Zδ is defined by (5.38) and where v satisfies (5.39). One formally
obtains
(5.41)
Z
Z
Z
A(x)DuDuZδ0 (u)v +
F (x, u)Zδ (u)v =
Ω
Ω
A(x)DuDvZδ (u),
Ω
and one observes that, denoting by Yδ the primitive of the function Zδ
(see (5.43) below),
Z
Z


F (x, u)v ≤
F (x, u)Zδ (u)v,



{u≤δ}
Ω

Z





A(x)DuDuZδ0 (u)v ≤ 0,



ZΩ
Z
Z



t
A(x)DuDvZδ (u) =
A(x)DYδ (u)Dv =
A(x)DvDYδ (u) =
Ω
ΩZ
Ω

Z

X



=
ĝi D(ϕ̂i Yδ (u)) + fˆYδ (u) =



Ω

i∈I

! Ω

Z

XZ
X



Zδ (u)ĝi Du ϕ̂i ,
ĝi Dϕ̂i + fˆ Yδ (u) +
=

Ω
i∈I
i∈I
where one finally uses 0 ≤ Yδ (s) ≤
Ω
3
δ to formally obtain estimate
2
(5.40).
These formal computations will be made mathematically correct in
the proof below.
As a consequence of Proposition 5.9 we have:
Proposition 5.11. Assume that the matrix A and the function F satisfy (2.3), (2.4) and (2.5). Then for every u solution to problem (2.1)
in the sense of Definition 3.6 one has
Z
(5.42)
F (x, u)v = 0 ∀v ∈ V(Ω), v ≥ 0.
{u=0}
SEMILINEAR PROBLEMS WITH A STRONG SINGULARITY AT u = 0
33
Proof of Proposition 5.11.
Let δ > 0 tend to 0 in (5.40). In the left-hand side one has
Z
Z
F (x, u)v ≤
F (x, u)v,
{u=0}
{u≤δ}
while in the right-hand side
Zδ (u) → χ{u=0} a.e. x ∈ Ω;
1
but since u ∈ Hloc
(Ω), one has
Du = 0 a.e. x ∈ {u = 0},
and therefore, since gˆi Du ϕ̂i ∈ L1 (Ω), and since 0 ≤ Zδ(u) ≤ 1, one has,
by Lebesgue’s dominated convergence Theorem,
Z
Zδ (u)gˆi Du ϕ̂i → 0 as δ → 0.
Ω
This proves (5.42).
Proof of Proposition 5.9.
In addition to the function Zδ (s) defined by (5.38), we define for
δ>0


if 0 ≤ s ≤ δ,
s
s2
δ
(5.43)
Yδ (s) = − 2δ + 2s − 2 if δ ≤ s ≤ 2δ,

3δ
if 2δ ≤ s,
2


if 0 ≤ s ≤ δ,
0
s2
δ
(5.44)
Rδ (s) = 2δ − 2 if δ ≤ s ≤ 2δ,

3δ
if 2δ ≤ s.
2
Observe that Zδ , Yδ and Rδ are Lipschitz continuous and piecewise C 1
functions with
(5.45)


Yδ0 (s) = Zδ (s), Zδ0 (s) = − 1 χ
(s), Rδ0 (s) = −sZδ0 (s),
{δ<s<2δ}
δ
3

Yδ (s) = sZδ (s) + Rδ (s), 0 ≤ Yδ (s) ≤ δ, ∀s ≥ 0.
2
First step. In this first step we will prove that for every δ > 0
(
1
Yδ (u) ∈ Hloc
(Ω) ∩ L∞ (Ω),
(5.46)
ϕYδ (u) ∈ H01 (Ω) ∀ϕ ∈ H01 (Ω) ∩ L∞ (Ω),
(5.47)
Zδ (u)v ∈ V(Ω) ∀v ∈ V(Ω),
34
D. GIACHETTI, P.J. MARTÍNEZ-APARICIO, AND F. MURAT
(5.48)
Rδ (u) ∈ H01 (Ω) ∩ L∞ (Ω).
1
1
(Ω) ∩ L∞ (Ω) with
(Ω) by (3.10 i), one has Yδ (u) ∈ Hloc
Since u ∈ Hloc
DYδ (u) = Yδ0 (u)Du = Zδ (u)Du in D0 (Ω),
1
and for every ϕ ∈ H01 (Ω) ∩ L∞ (Ω), one has ϕYδ (u) ∈ Hloc
(Ω) ∩ L∞ (Ω)
with
D(ϕYδ (u)) = ϕZδ (u)Du + Yδ (u)Dϕ in D0 (Ω),
which using (3.16) and the last assertion of (5.45) implies that
D(ϕYδ (u)) ∈ (L2 (Ω))N , and therefore that ϕYδ (u) ∈ H 1 (Ω) ∩ L∞ (Ω).
Since
3
0 ≤ ϕYδ (u) ≤ δϕ,
2
1
and since ϕ ∈ H0 (Ω), Lemma A.1 of Appendix A below completes the
proof of (5.46).
1
On the other hand Zδ (u) ∈ Hloc
(Ω) and
1
DZδ (u) = − χ{δ<u<2δ} Du in D0 (Ω).
δ
In view of (3.10 iii) with k = δ, this implies that DZδ (u) ∈ (L2 (Ω))N ,
and therefore
(5.49)
Zδ (u) ∈ H 1 (Ω) ∩ L∞ (Ω),
which in turn implies that
(5.50)
Zδ (u)ϕ ∈ H01 (Ω) ∩ L∞ (Ω) ∀ϕ ∈ H01 (Ω) ∩ L∞ (Ω).
Consider now v ∈ V(Ω) which satisfies (5.39).
Observe that
1
∞
Zδ (u)v ∈ H0 (Ω) ∩ L (Ω), and that

−div tA(x)D(Zδ (u)v) =



t
t


= Zδ (u)(−div A(x)Dv) − A(x)DvDZδ (u) +
(5.51)
+v(−div tA(x)DZδ (u)) − tA(x)DZδ (u)Dv =

P


= i∈I Zδ (u)ϕ̂i (−div ĝi ) + Zδ (u)fˆ − tA(x)DvDZδ (u) +



+v(−div tA(x)DZδ (u)) − tA(x)DZδ (u)Dv in D0 (Ω).
This proves that Zδ (u)v ∈ V(Ω), namely (5.47).
1
Finally, since u ∈ Hloc
(Ω) by (3.10 i), one has
1
Rδ (u) ∈ Hloc
(Ω) ∩ L∞ (Ω)
SEMILINEAR PROBLEMS WITH A STRONG SINGULARITY AT u = 0
35
with
u
χ{δ<u<2δ} Du in D0 (Ω),
δ
and therefore in view of (3.10 iii) with k = δ, DRδ (u) ∈ (L2 (Ω))N ,
which proves that Rδ (u) ∈ H 1 (Ω) ∩ L∞ (Ω). Then using the inequality
DRδ (u) = Rδ0 (u)Du =
0 ≤ Rδ (u) ≤ G δ (u),
2
δ
2
property (3.10 iii) with k = and Lemma A.1 of Appendix A below
completes the proof of (5.48).
Second step. In this step we fix δ > 0 and k such that
k ≥ 2δ.
(5.52)
Since Zδ (u)v ∈ V(Ω) for every v ∈ V(Ω) (see (5.47)) with Zδ (u)v ≥ 0
when v ≥ 0, we can take Zδ (u)v as test function in (3.11 ii), obtaining
in view of (5.51)
(5.53)Z

XZ
t


gˆi D(Zδ (u)ϕ̂i Tk (u)) +
A(x)D(Zδ (u)v)DGk (u) +



Ω
Ω
i∈I

Z
Z



t


+ Zδ (u)fˆ Tk (u) − A(x)DvDZδ (u) Tk (u) +



Z
Ω
 ZΩ
t
t
A(x)DZδ (u)Dv Tk (u) =
A(x)DZδ (u)D(vTk (u)) −
+

Ω
Ω



= h−div tA(x)D(Zδ (u)v), Gk (u)iH −1 (Ω),H01 (Ω) +



+hh−div tA(x)D(Z (u)v), T (u)ii =

δ
k
Ω

Z




=
F (x, u)Zδ (u)v.
Ω
As far as the first term of the left-hand side of (5.53) is concerned,
we have, since k ≥ 2δ,
Z
t
(5.54)
A(x)D(Zδ (u)v)DGk (u) = 0;
Ω
1
Hloc (Ω)
indeed since u ∈
by (3.10 i) one has

tA(x)D(Zδ (u)v)DGk (u) =
1
= −tA(x)DuDu χ{δ<u<2δ} χ{u>k} + tA(x)DvDu Zδ (u)χ{u>k} in D0 (Ω),
δ
where each term is zero almost everywhere when k ≥ 2δ.
As far as the fourth term of the left-hand side of (5.53) is concerned,
1
we have, since u ∈ Hloc
(Ω) and since k ≥ 2δ, using (5.45),
DZδ (u) Tk (u) = Du Zδ0 (u) Tk (u) = −Du Rδ0 (u) = −DRδ (u) in D0 (Ω),
36
D. GIACHETTI, P.J. MARTÍNEZ-APARICIO, AND F. MURAT
which using the fact that Rδ (u) ∈ H01 (Ω) (see (5.48)) and (5.39) implies
that
Z
 Z
t
t

−
A(x)DvDZδ (u) Tk (u) =
A(x)DvDRδ (u) =



Ω
Ω

t
= h−div
(5.55)
Z A(x)Dv, Rδ (u)iHZ−1 (Ω),H01 (Ω) =

X



gˆi D(ϕ̂i Rδ (u)) + fˆRδ (u).
=
i∈I
Ω
Ω
As far as the fifth term of the left-hand side of (5.53) is concerned,
1
(Ω),
we have, since u ∈ Hloc
(
t
A(x)DZδ (u)D(vTk (u)) =
= tA(x)DZδ (u)Dv Tk (u) + tA(x)DZδ (u)DTk (u) v in D0 (Ω),
which since k ≥ 2δ, implies that
Z


 tA(x)DZδ (u)D(vTk (u)) =
ΩZ
Z
(5.56)
1
t
t


A(x)DZδ (u)Dv Tk (u) −
A(x)DuDu v.
=
δ {δ<u<2δ}
Ω
Collecting together (5.53), (5.54), (5.55) and (5.56), we have proved
that
(5.57)
Z
XZ

ˆ Zδ (u)Tk (u) + Rδ (u) =

g
ˆ
D(
ϕ̂
Z
(u)T
(u)
+
R
(u)
)
+
f
i
i
δ
k
δ

Ω
i∈IZ Ω
Z
1

t

A(x)DuDu v.
F (x, u)Zδ (u)v +
=
δ {δ<u<2δ}
Ω
Since when k ≥ 2δ one has (see (5.45))

Zδ (u) ≥ χ{u≤δ} , tA(x)DuDu v ≥ 0,




Zδ (u)Tk (u) + Rδ (u) = Zδ (u)u + Rδ (u) = Yδ (u),
gˆi D(ϕ̂i Yδ (u)) = gˆi Dϕ̂i Yδ (u) + gˆi Du Zδ (u)ϕ̂i ,




0 ≤ Yδ (u) ≤ 3 δ,
2
estimate (5.40) follows from (5.57).
Proposition 5.9 is proved.
5.4. A priori estimate of β(Tk (u)) in H01 (Ω)
This fourth a priori estimate is actually in some sense a regularity
result, since it asserts that for every u solution to problem (2.1) in the
sense of Definition 3.6, a certain function of Tk (u) actually belongs to
H01 (Ω). This property will be used in the proofs of the Comparison
SEMILINEAR PROBLEMS WITH A STRONG SINGULARITY AT u = 0
37
Principle 7.1, of the Uniqueness Theorem 4.3, of the Equivalence result
of Proposition B.1, and of the Equivalence result of Proposition B.2 of
Appendix B below.
Define the function β : s ∈ [0, +∞[→ β(s) ∈ [0, +∞[ by
Z sp
(5.58)
β(s) =
Γ0 (t)dt.
0
Proposition 5.12. Assume that the matrix A and the function F satisfy (2.3), (2.4) and (2.5). Then for every u solution to problem (2.1)
in the sense of Definition 3.6 one has
(5.59)
β(Tk (u)) ∈ H01 (Ω) ∀k > 0,
with
(5.60)
αkDβ(Tk (u))k2(L2 (Ω))N ≤ khkL1 (Ω) ∀k > 0.
Remark 5.13. (Formal proof of Proposition 5.12). Estimate
(5.60) can be obtained formally by taking Γ(Tk (u)) as test function
in equation (2.1), using the coercivity (2.3) and the growth condition
(2.5 iii), which implies, since Γ is increasing, that
0 ≤ F (x, u)Γ(Tk (u)) ≤
h(x)
Γ(Tk (u)) ≤ h(x).
Γ(u)
This formal computation will be made mathematically correct in the
proof below.
Proof of Proposition 5.12.
As in (5.5), we define, for every δ and k with 0 < δ < k, the function
Sδ,k as


if 0 ≤ s ≤ δ,
0
(5.61)
Sδ,k (s) = s − δ if δ ≤ s ≤ k,

k − δ if k ≤ s.
As in the beginning of the proof of Proposition 5.1, one can prove
that the function Γ(Sδ,k (u)) belongs to V(Ω) and that (compare with
(5.10))
−div tA(x)DΓ(Sδ,k (u)) = ψδ (u)(−div tA(x)DΓ(Sδ,k (u)) in D0 (Ω),
where ψδ (s) is a C 1 nondecreasing function such that
ψδ (s) = 0 for 0 ≤ s ≤
δ
and ψ(s) = 1 for s ≥ δ.
2
38
D. GIACHETTI, P.J. MARTÍNEZ-APARICIO, AND F. MURAT
Since Γ(Sδ,k (u)) ∈ V(Ω) and since Γ(Sδ,k (u)) ≥ 0, we can use
Γ(Sδ,k (u)) as test function in (3.11 ii). We get
Z
Z
t

 A(x)DΓ(Sδ,k (u))DGk (u) + tA(x)DΓ(Sδ,k (u))D(ψδ (u)Tk (u)) =



Ω
Ω

= h−div tA(x)DΓ(S (u)), G (u)i −1
1
+
δ,k
k
H
(Ω),H0 (Ω)
t

+hh−div
A(x)DΓ(Sδ,k (u)), Tk (u)iiΩ =

Z




=
F (x, u)Γ(Sδ,k (u)),
Ω
which can be written as
(5.62)
Z
Z
t

 A(x)DΓ(Sδ,k (u))DGk (u) + tA(x)DΓ(Sδ,k (u))Dψδ (u) Tk (u) +
Ω
Z
ΩZ

t
+
A(x)DΓ(Sδ,k (u))DTk (u) ψδ (u) =
F (x, u)Γ(Sδ,k (u)).
Ω
Ω
Note that the two first integrals are zero, since DΓ(Sδ,k (u)) is zero
outside of the set {δ < u < k}, while DGk (u) is zero outside of the set
{u > k} and Dψδ (u) is zero outside of the set {u < δ}.
Note also that ψδ (u) = 1 on the set {u ≥ δ} while DΓ(Sδ,k (u)) = 0
outside of this set. Therefore the third term of (5.62) can be written
as
Z
t


A(x)DΓ(Sδ,k (u))DTk (u) ψδ (u) =



Ω
Z



t

A(x)DSδ,k (u)DTk (u)Γ0 (Sδ,k (u)) =
=
ZΩ
(5.63)

t

A(x)DSδ,k (u)DSδ,k (u)Γ0 (Sδ,k (u)) ≥
=



ΩZ
Z




2
0
≥ α |DSδ,k (u)| Γ (Sδ,k (u)) = α |Dβ(Sδ,k (u))|2 .
Ω
Ω
For what concerns the right-hand side of (5.62), we have, using the
growth condition (2.5 iii) and the fact that Γ is increasing,
(5.64)
F (x, u)Γ(Sδ,k (u)) ≤
h(x)
Γ(Sδ,k (u)) ≤ h(x).
Γ(u)
By (5.62), (5.63) and (5.64) we get
Z
(5.65)
α |Dβ(Sδ,k (u))|2 ≤ khkL1 (Ω) ∀δ, 0 < δ < k.
Ω
Passing to the limit as δ tends to zero gives estimate (5.60).
SEMILINEAR PROBLEMS WITH A STRONG SINGULARITY AT u = 0
39
On the other hand, the fact that for every δ with 0 < δ < k
(
|Dβ(Sδ,k (u))| = |β 0 (Sδ,k (u))χ{δ<u<k} Du| ≤ β 0 (k)|DGδ (u)|,
0 ≤ β(Sδ,k (u)) ≤ β(Gδ (u)),
condition (3.10 iii) and Lemma A.1 of Appendix A below imply that
β(Sδ,k (u)) ∈ H01 (Ω). The convergence
β(Sδ,k (u)) * β(Tk (u)) in H 1 (Ω) weakly as δ → 0,
then implies that β(Tk (u)) ∈ H01 (Ω), i.e. the regularity result (5.59).
6. Proofs of the Stability Theorem 4.2
and of the Existence Theorem 4.1
6.1. Proof of the Stability Theorem 4.2
First step. Since for every n the function Fn (x, s) satisfies assumptions (2.4) and (2.5) for the same h and the same Γ, every solution
un to problem (2.1)n in the sense of Definition 3.6 satisfies the a priori
estimates (5.1), (5.3), (5.16), (5.17), (5.18), (5.19) and (5.40).
1
Therefore un is bounded in L2 (Ω) and in Hloc
(Ω), and there exist a
subsequence, still labelled by n, and a function u∞ such that
(6.1)

1
un * u∞ in L2 (Ω) weakly, in Hloc
(Ω) weakly and a.e. in Ω,



G (u ) * G (u ) in H 1 (Ω) weakly ∀k > 0,
k n
k ∞
0

ϕTk (un ) * ϕTk (u∞ ) in H01 (Ω) weakly ∀k > 0,



∀ϕ ∈ H01 (Ω) ∩ L∞ (Ω).
Since un ≥ 0, this function u∞ satisfies u∞ ≥ 0. In conclusion u∞
satisfies (3.10).
Fix now a function v
X
v ∈ V(Ω), v ≥ 0, with − div tA(x)Dv =
ϕ̂i (−div ĝi ) + fˆ in D0 (Ω).
i∈I
Using v as test function in (3.11 ii)n , we obtain
(6.2)
Z
Z
XZ
t

A(x)DvDGk (un ) +
gˆi D(ϕ̂i Tk (un )) + fˆTk (un ) =



Ω
 Ω
i∈I Ω
t
t
= h−div
A(x)Dv, Gk (un )iH −1 (Ω),H01 (Ω) + hh−div A(x)Dv, Tk (un )iiΩ =

Z



=
F (x, u )v.
n
Ω
n
40
D. GIACHETTI, P.J. MARTÍNEZ-APARICIO, AND F. MURAT
Since the left-hand side of (6.2) is bounded in n for every k > 0 fixed
in view of the estimates (5.1) and (5.16), we have
Z
Fn (x, un )v ≤ C(v) < +∞ ∀n,
Ω
which using the almost everywhere convergence of un to u∞ , assumption (4.1) on the functions Fn and Fatou’s Lemma gives
Z
F∞ (x, u∞ )v ≤ C(v) < +∞ ∀n,
(6.3)
Ω
namely (3.11 i)∞ .
It remains to prove that (3.11 ii)∞ holds true and that the convergences in (6.1) are strong.
Second step. For δ > 0 fixed, we write (6.2) as
(6.4)Z




XZ
t
A(x)DvDGk (un ) +
Ω
Z
Zi∈I


Fn (x, un )v +
=
{un ≤δ}
Z
gˆi D(ϕ̂i Tk (un )) +
fˆTk (un ) =
Ω
Ω
Fn (x, un )v.
{un >δ}
It is easy to pass to the limit in the left-hand side of (6.4), obtaining
(6.5)
Z
Z
XZ
t


gˆi D(ϕ̂i Tk (un )) + fˆTk (un ) →
A(x)DvDGk (un ) +



Ω
Ω
Ω
 Z
i∈I
Z
XZ
t
→
A(x)DvDGk (u∞ ) +
gˆi D(ϕ̂i Tk (u∞ )) + fˆTk (u∞ )



Ω
Ω
Ω

i∈I


as n → +∞.
Concerning the first term of the right-hand side of (6.4) we use the
a priori estimate (5.40), namely
Z




∀δ > 0, {un ≤δ} Fn (x, un )v ≤
!
Z X
XZ
3

ˆ

gˆi Dϕ̂i + f δ +
Zδ (un )gˆi Dun ϕ̂i ,

≤ 2
Ω
Ω
i∈I
i∈I
in which we pass to the limit in n for δ > 0 fixed. Since Zδ (un )gˆi tends
strongly to Zδ (u∞ )gˆi in (L2 (Ω))N while Dun ϕ̂i tends weakly to Du∞ ϕ̂i
SEMILINEAR PROBLEMS WITH A STRONG SINGULARITY AT u = 0
41
in (L2 (Ω))N (see (5.18)), we obtain
Z


∀δ > 0, lim sup
Fn (x, un )v ≤


n
{un ≤δ} !
Z X
(6.6)
XZ
3


Zδ (u∞ )gˆi Du∞ ϕ̂i .
gˆi Dϕ̂i + fˆ δ +
≤

2
Ω
Ω
i∈I
i∈I
Since
Zδ (u∞ ) → χ{u∞ =0} a.e. x ∈ Ω, as δ → 0,
1
(Ω) implies that Du∞ = 0 almost everywhere on
and since u∞ ∈ Hloc
{x ∈ Ω : u∞ (x) = 0}, the right-hand side of (6.6) tends to 0 as δ tends
to 0.
We have proved that the first term of the right-hand side of (6.4)
satisfies
Z
Fn (x, un )v → 0 as δ → 0.
(6.7)
lim sup
n
{un ≤δ}
Third step. Let us now prove that
Z
F∞ (x, u∞ )v = 0.
(6.8)
{u∞ =0}
For that we observe that for every δ > 0
Z
Z


Fn (x, un )χ{un ≤δ} v =
Fn (x, un )v ≤

{uZ∞ =0}
{u∞ =0}∩{un ≤δ}
(6.9)


Fn (x, un )v.
≤
{un ≤δ}
Since un converges almost everywhere to u∞ , one has, for every δ > 0
fixed
χ{un ≤δ} → 1 a.e. x ∈ {x ∈ Ω : u∞ (x) = 0},
while in view of assumption (4.1), one has
Fn (x, un (x)) → F∞ (x, u∞ (x)) a.e. x ∈ Ω.
Applying Fatou’s Lemma to the left-hand side of (6.9), one obtains
Z
Z
F∞ (x, u∞ )v ≤ lim sup
Fn (x, un ) v ∀δ > 0,
{u∞ =0}
n
which in view of (6.7) implies (6.8).
{un ≤δ}
42
D. GIACHETTI, P.J. MARTÍNEZ-APARICIO, AND F. MURAT
Fourth step. We finally pass to the limit in n for δ > 0 fixed in the
second term of the right-hand side of (6.4), namely in
Z
Z
(6.10)
Fn (x, un )v =
Fn (x, un )χ{un >δ} v.
{un >δ}
Ω
We define C as the set of the (exceptional) values of δ > 0 such that
δ ∈ C ⇐⇒ meas{x ∈ Ω : u∞ (x) = δ} > 0.
(6.11)
The set C is at most countable, and we have
∀δ 6∈ C, meas{x ∈ Ω : u∞ (x) = δ} = 0.
(6.12)
For every fixed δ > 0 we have, in view of (2.5 iii) and (2.5 ii),
0 ≤ Fn (x, un )χ{un >δ} v ≤
h(x)
h(x)
χ{un >δ} v ≤
v a.e. x ∈ Ω,
Γ(un )
Γ(δ)
and we have, in view of the almost everywhere convergence of un to
u∞ and of the convergence (4.1) of Fn to F∞ ,
Fn (x, un )v → F∞ (x, u∞ )v a.e. x ∈ Ω,
χ{un >δ} → χ{u∞ >δ} a.e. x ∈ Ω \ {x ∈ Ω : u∞ (x) = δ}.
Choosing δ > 0 with δ 6∈ C (see (6.12)), Lebesgue’s dominated convergence Theorem implies that

∀δ > 0, δ 6∈ C,
Z
Z
(6.13)
F∞ (x, u∞ )v as n → +∞.
Fn (x, un )v →

{u∞ >δ}
{un >δ}
On the other hand, using the fact that
χ{u∞ >δ} → χ{u∞ >0} a.e. x ∈ Ω as δ → 0,
the fact that F∞ (x, u∞ )v belongs to L1 (Ω) (see (6.3)), and finally (6.8),
we have
(6.14)
Z
Z
Z

F∞ (x, u∞ )v →
F∞ (x, u∞ )v =
F∞ (x, u∞ )v
{u∞ >δ}

{u∞ >0}
Ω
as δ → 0.
Collecting the results obtained in (6.13) and (6.14), we have proved
that the second term of the right-hand side of (6.4) satisfies
Z
Z
(6.15)
lim
Fn (x, un )v →
F∞ (x, u∞ )v as δ → 0, δ 6∈ C.
n
{un >δ}
Ω
SEMILINEAR PROBLEMS WITH A STRONG SINGULARITY AT u = 0
43
Fifth step. Collecting the results obtained in (6.5), (6.7) and (6.15),
we pass to the limit in each term of (6.4), first in n for δ > 0 fixed, and
then in δ. This proves that for every v ∈ V(Ω), v ≥ 0
Z
Z
XZ
t


A(x)DvDGk (u∞ ) +
gˆi D(ϕ̂i Tk (u∞ )) + fˆTk (u∞ ) =

Ω
Ω
i∈I Ω
Z


=
F∞ (x, u∞ )v,
Ω
which is nothing but (3.11 ii)∞ .
We have proved that u∞ is a solution to problem (2.1)∞ in the sense
of Definition 3.6.
It only remains to prove that the convergences in (6.1) are strong
(see (4.2)). To this aim it is sufficient to prove the following two strong
convergences
Gk (un ) → Gk (u∞ ) in H01 (Ω) strongly ∀k > 0,
(6.16)
(6.17)
(
ϕDTk (un ) → ϕDTk (u∞ ) in (L2 (Ω))N strongly ∀k > 0,
∀ϕ ∈ H01 (Ω) ∩ L∞ (Ω).
Indeed one has un = Tk (un ) + Gk (un ), and the strong convergence
of Tk (un ) in L2 (Ω) follows from Lebesgue’s dominated convergence
Theorem.
Sixth step. As far as (6.16) is concerned, the strong convergence
follows from the energy equality (5.4)n , namely the fact that
Z
Z
(6.18)
A(x)DGk (un )DGk (un ) =
Fn (x, un )Gk (un ) ∀k > 0.
Ω
Ω
It is easy to pass to the limit in the right-hand side of (6.18). Indeed
the inequality
0 ≤ Fn (x, un )Gk (un ) ≤
h(x)
Gk (un )
Γ(k)
and the boundedness of Gk (un ) in H01 (Ω) (see (5.1)) imply that the
sequence Fn (x, un )Gk (un ) is equintegrable in n; since
Fn (x, un )Gk (un ) → F∞ (x, u∞ )Gk (u∞ ) a.e. x ∈ Ω,
using Vitali’s Theorem one obtains for every k > 0 fixed
Z
Z
(6.19)
Fn (x, un )Gk (un ) →
F∞ (x, u∞ )Gk (u∞ ) as n → +∞.
Ω
Ω
44
D. GIACHETTI, P.J. MARTÍNEZ-APARICIO, AND F. MURAT
On the other hand, the energy equality (5.4)∞ asserts that
Z
Z
(6.20)
A(x)DGk (u∞ )DGk (u∞ ) =
F∞ (x, u∞ )Gk (u∞ ) ∀k > 0.
Ω
Ω
Collecting together (6.18), (6.19) and (6.20), one has for every k > 0
fixed
Z
Z
A(x)DGk (un )DGk (un ) →
A(x)DGk (u∞ )DGk (u∞ ) as n → +∞,
Ω
Ω
which, with the weak convergence in (L2 (Ω))N of DGk (un ) to DGk (u∞ )
implies the strong convergence (6.16).
Seventh step. As far as (6.17) is concerned, writing equation (5.29)
for F = Fn and u = un , one obtains
(6.21)
Z


A(x)DTk (un )DTk (un ) ϕ2 =



Z
 Ω Z
= −2 A(x)DTk (un )Dϕ ϕTk (un ) − 2 A(x)DGk (un )Dϕ ϕTk (un )+

Z
Ω
Ω




+ Fn (x, un )ϕ2 Tk (un ) ∀ϕ ∈ H01 (Ω) ∩ L∞ (Ω).
Ω
Let us pass to the limit in the right-hand side of (6.21) as n tends
to +∞. In view of (6.1) one has
(6.22)

Z
Z


−2 A(x)DTk (un )Dϕ ϕTk (un ) − 2 A(x)DGk (un )Dϕ ϕTk (un ) →



ΩZ
Ω Z
→ −2 A(x)DTk (u∞ )Dϕ ϕTk (u∞ ) − 2



Ω

as n → +∞.
A(x)DGk (u∞ )Dϕ ϕTk (u∞ )
Ω
As far as the last integral of the right-hand side of (6.21) is concerned,
we write, for every fixed δ > 0,
Z


 Fn (x, un )ϕ2 Tk (un ) =
ΩZ
Z
(6.23)
2


Fn (x, un )ϕ Tk (un ) +
Fn (x, un )ϕ2 Tk (un ).
=
{un ≤δ}
{un >δ}
For the first term of the right-hand side of (6.23), we use the a priori
estimate (5.40) with v = ϕ2 ; indeed ϕ2 ∈ V(Ω) (see (3.8)) with
−div tA(x)Dϕ2 = ϕ̂(−div ĝ) + fˆ in D0 (Ω),
SEMILINEAR PROBLEMS WITH A STRONG SINGULARITY AT u = 0
45
with ϕ̂ = 2ϕ, ĝ = tA(x)Dϕ and fˆ = −2 tA(x)DϕDϕ; this yields, since
ĝDϕ̂ + fˆ = 0,
(6.24) Z
Z

2


0≤
Fn (x, un )ϕ Tk (un ) ≤ k
Fn (x, un )ϕ2 ≤



{un ≤δ}

Z{un ≤δ}

≤ 2k Zδ (un ) tA(x)DϕDun ϕ =


ZΩ
Z




t
= 2k Zδ (un ) A(x)DϕDTk (un ) ϕ + 2k Zδ (un ) tA(x)DϕDGk (un ) ϕ.
Ω
Ω
We pass to the limit in the right-hand side of (6.24) first for δ > 0
fixed as n tends to +∞ thanks to (6.1), and then as δ tends to 0.
Since Zδ (u∞ ) tends to χ{u∞ =0} in L∞ (Ω) weak-star and since Du∞ = 0
almost everywhere in {u∞ = 0}, we obtain that
Z
(6.25)
lim sup
Fn (x, un )ϕ2 Tk (un ) → 0 as δ → 0,
n
{un ≤δ}
For the second term of the right-hand side of (6.23), we repeat the
proof that we performed above to prove (6.15), and we obtain that
Z
Z

2
lim
F∞ (x, u∞ )ϕ2 Tk (u∞ )
Fn (x, un )ϕ Tk (un ) →
n
(6.26)
Ω
{un >δ}

as δ → 0, δ 6∈ C.
On the other hand, writing equation (5.29) for F = F∞ and u = u∞ ,
one has
(6.27)
Z


A(x)DTk (u∞ )DTk (u∞ ) ϕ2 =



Z
 Ω Z
= −2 A(x)DTk (u∞ )Dϕ ϕTk (u∞ ) − 2 A(x)DGk (u∞ )Dϕ ϕTk (u∞ ) +

Z
Ω
Ω



2
1

+ F∞ (x, u∞ )ϕ Tk (u∞ ) ∀ϕ ∈ H0 (Ω) ∩ L∞ (Ω).
Ω
Collecting together (6.21), (6.22), (6.23), (6.25), (6.26) and (6.27)
one has for every k > 0 fixed
Z
Z
 A(x)DT (u )DT (u ) ϕ2 →
A(x)DTk (u∞ )DTk (u∞ ) ϕ2
k n
k n
Ω
 Ω
as n → +∞,
which, with the weak convergence in (L2 (Ω))N of ϕDTk (un ) to
ϕDTk (u∞ ) implies the strong convergence (6.17).
46
D. GIACHETTI, P.J. MARTÍNEZ-APARICIO, AND F. MURAT
This completes the proof of the Stability Theorem 4.2.
6.2. Proof of the Existence Theorem 4.1
Consider the problem
(
un ∈ H01 (Ω),
(6.28)
0
−div A(x)Dun = Tn (F (x, u+
n )) in D (Ω),
where Tn is the truncation at height n.
Since
Tn (F (x, s+ )) : (x, s) ∈ Ω×] − ∞, +∞[→ Tn (F (x, s+ )) ∈ [0, +∞[
is a Carathéodory function which is almost everywhere bounded by
n, Schauder’s fixed point theorem implies that problem (6.28) has at
least a solution. Moreover, since Tn (F (x, s+ )) ≥ 0, this solution is
nonnegative by the weak maximum principle, so that un ≥ 0, and
Tn (F (x, u+
n )) = Tn (F (x, un )).
Define now the function Fn by
Fn (x, s) = Tn (F (x, s)) a.e. x ∈ Ω, ∀s ∈ [0, +∞[.
For every given n, the function Fn (which is bounded) satisfies the
growth condition (B.1) of Appendix B below, and every un solution to
(6.28) satisfies (B.2) and (B.3). Then the equivalence result of Proposition B.1 implies that un is actually a solution to problem (2.1)n in the
sense of Definition 3.6, where (2.1)n is the problem (2.1) where F (x, s)
is replaced by Tn (F (x, s)).
It is clear that the functions Fn satisfy (2.4) and (2.5) with the
functions h and Γ which appear in the definition of the function F .
Moreover it is not difficult to verify that the function Fn satisfy (4.1)
with F∞ given by
F∞ (x, s) = F (x, s).
The Stability Theorem 4.2 then implies that there exists a subsequence of un whose limit u∞ is a solution to problem (2.1) in the sense
of Definition 3.6.
This proves the Existence Theorem 4.1.
7. Comparison Principle
and proof of the Uniqueness Theorem 4.3
In this section we prove a comparison result which uses assumption
(2.6), namely the fact that F (x, s) is nonincreasing with respect to s.
Note that we never used this assumption in the rest of the present
paper (except as far as the Uniqueness Theorem 4.3 is concerned).
SEMILINEAR PROBLEMS WITH A STRONG SINGULARITY AT u = 0
47
Proposition 7.1. (Comparison Principle). Assume that the matrix
A satisfies (2.3). Let F1 (x, s) and F2 (x, s) be two functions satisfying
(2.4) and (2.5) (possibly for different functions h and Γ). Assume
moreover that
(7.1) either F1 (x, s) or F2 (x, s) is nonincreasing, i.e. satisfies (2.6),
and that
(7.2)
F1 (x, s) ≤ F2 (x, s) a.e. x ∈ Ω,
∀s ≥ 0.
Let u1 and u2 be any solutions in the sense of Definition 3.6 to problems (2.1)1 and (2.1)2 , where (2.1)1 and (2.1)2 are (2.1) with F (x, s)
replaced respectively by F1 (x, s) and F2 (x, s). Then one has
u1 (x) ≤ u2 (x) a.e. x ∈ Ω.
(7.3)
Remark 7.2. The proof of the Comparison Principle of Proposition 7.1
is based on the use of the test function (B1 (Tk+ (u1 − u2 )))2 . A similar
test function has been used by L. Boccardo and J. Casado-Dı́az in
[1] to prove the uniqueness of solution to problem (2.1) obtained by
approximation.
Note that the use of this test function is allowed by the regularity
property (5.59) proved in Proposition 5.12.
Proof of the Uniqueness Theorem 4.3.
Applying the Comparison Principle to the case where
F1 (x, s) = F2 (x, s) = F (x, s) with F (x, s) satisfying (2.6) immediately
proves the Uniqueness Theorem 4.3.
Proof of Proposition 7.1.
First step. Let k > 0 be fixed. Define ϕ by
ϕ = B1 (Tk+ (u1 − u2 )),
(7.4)
where B1 : s ∈ [0, +∞[→ B1 (s) ∈ [0, +∞[ is the function defined by
Z s
B1 (s) =
β1 (t)dt ∀s ≥ 0,
0
Z sp
where β1 (s) =
Γ01 (t)dt and Γ1 is the function for which F1 satisfies
0
(2.5).
In this step we will prove that
(7.5)
ϕ ∈ H01 (Ω) ∩ L∞ (Ω).
1
Since u1 and u2 belong to Hloc
(Ω), and since β1 ∈ C 0 ([0, +∞[), the
1
function ϕ belongs to Hloc
(Ω) and one has
(7.6)
Dϕ = β1 (Tk+ (u1 − u2 ))χ{0<u1 −u2 <k} (Du1 − Du2 ) in D0 (Ω),
48
D. GIACHETTI, P.J. MARTÍNEZ-APARICIO, AND F. MURAT
which, since Tk+ (s1 − s2 ) ≤ Tk (s1 ) for s1 ≥ 0 and s2 ≥ 0 and since β1
is nondecreasing, implies that
|Dϕ| ≤ β1 (Tk (u1 ))(|Du1 | + |Du2 |).
But in view of (5.59), β1 (Tk (u1 )) belongs to H01 (Ω) ∩ L∞ (Ω), and then
properties (3.10 iv) for u1 and u2 imply that Dϕ ∈ (L2 (Ω))N , and
therefore that ϕ ∈ H 1 (Ω).
Since B1 is nondecreasing one has 0 ≤ ϕ ≤ B1 (Tk (u1 )). We now
claim that
B1 (Tk (u1 )) ∈ H01 (Ω),
(7.7)
which by Lemma A.1 of Appendix A below completes the proof of
(7.5).
Let us now prove (7.7). As in the end of the proof of Proposition 5.12,
we observe that for every δ with 0 < δ < k and for the function Sδ,k
defined by (5.61) one has
(
|DB1 (Sδ,k (u))| = |β1 (Sδ,k (u))χ{δ<u<k} Du| ≤ β1 (k)|DGδ (u)|,
0 ≤ B1 (Sδ,k (u)) ≤ B1 (Gδ (u)).
Then Lemma A.1 implies that B1 (Sδ,k (u)) ∈ H01 (Ω). The convergence
B1 (Sδ,k (u)) * B1 (Tk (u)) in H 1 (Ω) weakly as δ → 0,
then implies (7.7).
Second step. Since ϕ2 ∈ V(Ω) in view of (7.5) and of Remark 3.5 ii),
we can take ϕ2 as test function in (3.11 ii)1 and (3.11 ii)2 . Taking the
difference of these two equations, we get
 Z


2 ϕ tA(x)DϕD(Gk (u1 ) − Gk (u2 )) +



ΩZ



t

+2
A(x)DϕD(ϕ(Tk (u1 ) − Tk (u2 ))) −



Ω

Z


t
−2
A(x)DϕDϕ (Tk (u1 ) − Tk (u2 )) =
(7.8)

Ω

= h−div tA(x)Dϕ2 , G (u ) − G (u )i −1


k 1
k 2 H (Ω),H01 (Ω) +


t
2


+hh−div
A(x)Dϕ , Tk (u1 ) − Tk (u2 )iiΩ =

Z




= (F1 (x, u1 ) − F2 (x, u2 )) ϕ2 .
Ω
Expanding in L1loc (Ω) the integrands of the three first lines of (7.8),
one realizes that their sum is nothing but 2ϕ tA(x)DϕD(u1 −u2 ), which
belongs to L1 (Ω) in view of Remark 3.9. Therefore one has
Z
Z
t
(7.9)
2 ϕ A(x)DϕD(u1 − u2 ) = (F1 (x, u1 ) − F2 (x, u2 )) ϕ2 ,
Ω
Ω
SEMILINEAR PROBLEMS WITH A STRONG SINGULARITY AT u = 0
49
which is formally easily obtained by taking ϕ2 as test function in (2.1)1
and (2.1)2 and making the difference.
Third step. Let us now prove that for ϕ given by (7.4) one has
(7.10)
(F1 (x, u1 ) − F2 (x, u2 )) ϕ2 ≤ 0 a.e. in x ∈ Ω.
Since F1 (x, u1 ) and F2 (x, u2 ) belong to L1 (Ω) (see (3.12)) and are
therefore finite almost everywhere, one has
(F1 (x, u1 ) − F2 (x, u2 )) ϕ2 = 0 on the set where ϕ2 = 0
(note that there is no indeterminacies of the types (∞ − ∞) and ∞ × 0
in the latest formula).
On the set where ϕ2 > 0, one has u1 > u2 . If F1 (x, s) is nonincreasing
with respect to s, one has, using this nonincreasing character and (7.2),
(
F1 (x, u1 ) − F2 (x, u2 ) ≤ F1 (x, u2 ) − F2 (x, u2 ) ≤ 0
on the set where ϕ2 > 0,
which implies
(F1 (x, u1 ) − F2 (x, u2 ))ϕ2 ≤ 0 on the set where ϕ2 > 0.
The case where F2 (x, s) is nonincreasing with respect to s is similar.
We have proved (7.10).
Fourh step. Collecting together (7.9), (7.10) and (7.6) gives
Z
2 B1 (Tk+ (u1 −u2 ))β1 (Tk+ (u1 −u2 )) tA(x)DTk+ (u1 −u2 )D(u1 −u2 ) ≤ 0.
Ω
Defining the function M1 by
Z sp
M1 (s) =
B1 (t)β1 (t)dt ∀s > 0,
0
this implies in view of the coercivity (2.3) of the matrix A that
DM1 (Tk+ (u1 − u2 )) = 0 on Ω. Therefore M1 (Tk+ (u1 − u2 )) is a constant in each connected component ω of Ω, and since the function M1
is strictly increasing, there exists a nonnegative constant Cω such that
Tk+ (u1 − u2 ) = Cω in ω, and therefore β1 (Tk+ (u1 − u2 )) = β1 (Cω ) in ω.
Since Tk+ (u1 − u2 ) ≤ Tk (u1 ) in Ω and since the function β1 is nondecreasing, one has
0 ≤ β1 (Cω ) = β1 (Tk+ (u1 − u2 )) ≤ β1 (Tk (u1 )) in ω.
By the regularity property (5.59), the function β1 (Tk (u1 )) belongs to
H01 (Ω) and therefore to H01 (ω) for every connected component ω of Ω.
By Lemma A.1 this implies that β1 (Cω ) ∈ H01 (Ω), and then that
Cω = 0. This proves that Tk+ (u1 − u2 ) = 0 in Ω, which implies (7.3)
since k > 0.
50
D. GIACHETTI, P.J. MARTÍNEZ-APARICIO, AND F. MURAT
8. The case with a zeroth-order term µu
with µ ∈ H −1 (Ω) ∩ M+
b (Ω)
In this section we consider the case where problem (1.1) is replaced
by problem (1.10), namely


in Ω,
u ≥ 0
(8.1)
−div A(x)Du + µu = F (x, u) in Ω,

u = 0
on ∂Ω,
where
(8.2)
µ ∈ H −1 (Ω) ∩ M+
b (Ω),
where M+
b (Ω) denotes the space of nonnegative bounded Radon measures in Ω.
In this section we present the variations which should be made in
the context of problem (8.1). For the comfort of the reader, we have
decided to treat this case in a separate section rather than writing
the whole of the present paper in this context, since the latest choice
would make the paper more difficult to read. Note that in this context the strong maximum principle does not hold true in general (see
Counterexample 8.3 below).
Let us note that problems like (8.1) naturally arise when performing
the homogenization of problem (1.1) (where there is no zeroth-order
term) posed in a domain Ω obtained from Ω by perforating Ω by small
holes, see our paper [10]; the appearance in Ω of the “strange term” µu
is then the “memory” of the Dirichlet homogeneous boundary condition
on ∂Ω .
8.1. Recalling the variational framework for the case with a
zeroth-order term µu with µ ∈ H −1 (Ω) ∩ M+
b (Ω)
Let us first recall the weak formulation of the problem (8.1) with a
given right-hand side
(8.3)
f ∈ L2 (Ω),
i.e. the case where F (x, s) = f (x) ∈ L2 (Ω), or in other terms the
problem of finding a function u which satisfies
(
−div A(x)Du + µu = f in Ω,
(8.4)
u=0
on ∂Ω,
where µ and f satisfy (8.2) and (8.3).
SEMILINEAR PROBLEMS WITH A STRONG SINGULARITY AT u = 0
51
(1)
1
When µ ∈ H −1 (Ω)∩M+
b (Ω) and when y ∈ H0 (Ω), it is well known
(see e.g. [6] Section 1 and [7] Section 2.2 for more details) that y (or
more exactly its quasi-continuous representative for the H01 (Ω) capacity) is µ-measurable and satisfies
Z
1
(8.5)
y ∈ L (Ω; dµ) and hµ, yiH −1 (Ω),H01 (Ω) =
ydµ;
Ω
1
moreover if z ∈ Hloc
(Ω) ∩ L∞ (Ω), then z is µ-measurable and satisfies
z ∈ L∞ (Ω; dµ) and kzkL∞ (Ω;dµ) = kzkL∞ (Ω) ;
(8.6)
it follows that when y ∈ H01 (Ω) ∩ L∞ (Ω), then y belongs to
L1 (Ω; dµ) ∩ L∞ (Ω; dµ), and therefore to Lp (Ω; dµ) for every p,
1 ≤ p ≤ +∞, and in particular to L2 (Ω; dµ).
Observing that H01 (Ω) ∩ L2 (Ω; dµ) is an Hilbert space, the weak
formulation of problem (8.4) is to find u such that
(8.7)

1
2
u
Z
Z dµ),
Z ∈ H0 (Ω) ∩ L (Ω;

Ω
fv
uvdµ =
A(x)DuDv +
Ω
∀v ∈ H01 (Ω) ∩ L2 (Ω; dµ).
Ω
This problem has a unique solution by Lax-Milgram Lemma.
8.2. The adaptation of Definition 3.6 to the case with a
zeroth-order term µu with µ ∈ H −1 (Ω) ∩ M+
b (Ω)
We can now present how Definition 3.6 should be adapted to the
case of problem (8.1).
Definition 8.1. Assume that the matrix A, the function F and the
Radon measure µ satisfy (2.3), (2.4), (2.5) and (8.2). We say that u is
a solution to problem (8.1) if u satisfies
(8.8)
(1)the

1
i) u ∈ L2 (Ω) ∩ Hloc
(Ω) ∩ L2 (Ω, dµ),



ii) u(x) ≥ 0 a.e. x ∈ Ω,

iii) Gk (u) ∈ H01 (Ω) ∀k > 0,



iv) ϕTk (u) ∈ H01 (Ω) ∀k > 0, ∀ϕ ∈ H01 (Ω) ∩ L∞ (Ω),
reader who would not like to use these results can continue reading the
present section just assuming that µ is an Lr (Ω) function, µ ≥ 0, with r as in (2.5)
and not only an element of H −1 (Ω) ∩ M+
b (Ω).
52
D. GIACHETTI, P.J. MARTÍNEZ-APARICIO, AND F. MURAT
(8.9)

∀v ∈ V(Ω) ∩ L2 (Ω, dµ), v ≥ 0,


X


t

with
−
div
A(x)Dv
=
ϕ̂i (−div ĝi ) + fˆ in D0 (Ω),




i∈I



∞
1

(Ω)
∩
L
(Ω), gˆi ∈ (L2 (Ω))N , f̂i ∈ L1 (Ω),
where
ϕ̂
∈
H
i

0



oneZ has





F (x, u)v < +∞,
i)
ZΩ
Z
XZ

t


ii)
A(x)DvDGk (u) +
gˆi D(ϕ̂i Tk (u)) + fˆTk (u) +



Ω
Ω
Ω

i∈I
 Z




+ uvdµ =



Ω


= h−div tA(x)Dv, Gk (u)iH −1 (Ω),H 1 (Ω) + hh−div tA(x)Dv, Tk (u)iiΩ +

Z
Z
0




+ uv dµ =
F (x, u)v ∀k > 0.
Ω
Ω
The differences between Definition 3.6 (where µ = 0) and Definition 8.1 consist only in the fact that u is assumed to belong to L2 (Ω, dµ)
in (8.8), that the test function v in Z(8.9) is also assumed to belong to
L2 (Ω, dµ) and that the new term
uvdµ appears in (8.9 ii); note
Ω
that this term has a meaning since u and v are assumed to belong to
L2 (Ω; dµ).
With this Definition, all the results and proofs of the present paper continue to hold true (once the necessary adaptations have been
made), the only exceptions being Proposition C.3 and Remark C.4
of Appendix C which are no more valid, since their proofs use the
strong maximum principle, which in general does not hold true for the
operator −div A(x)Du + µu (see Counterexample 8.3 below).
In particular, the analogous of Theorems 4.1 (Existence), 4.2 (Stability) and 4.3 (Uniqueness), as well as their proofs, hold true, once hypothesis (8.2) is made and Definition 3.6 is replaced by Definition 8.1.
The same holds true for the a priori estimates obtained in Section 5.
Let us just mention a few adaptations which have to be made in
Sections 5 and 6:
• in Proposition 5.1, estimate (5.1) has to be replaced by
Z
2
2
C 2 khkLr (Ω)
2
(8.10) kDGk (u)k(L2 (Ω))N +
uGk (u)dµ ≤ S2
∀k > 0;
α Ω
α Γ(k)2
SEMILINEAR PROBLEMS WITH A STRONG SINGULARITY AT u = 0
53
from (8.10) one deduces that
(8.11)
Z


u2 dµ =



Z

 ΩZ
2
(Tk (u))2 + (Gk (u))2 dµ ≤
= (Tk (u) + Gk (u)) dµ ≤ 2

Ω
Ω

Z
2


CS2 khkLr (Ω)

2
2

, ∀k > 0,
≤ 2k µ(Ω) + 2 u(Gk (u))dµ ≤ 2k µ(Ω) +
α Γ(k)2
Ω
which, taking k = k0 for some k0 > 0 fixed or minimizing the righthand side in k, provides an a priori estimate of kuk2L2 (Ω,dµ) which does
not depend on k;
• in Proposition 5.4, estimate (5.16) has to be replaced by
(8.12)
Z

2
2


u2 ϕ2 dµ ≤
kϕDTk (u)k(L2 (Ω))N +


α

{u<k}


2

 32k 2
C 2 khkLr (Ω)
≤ 2 kAk2(L∞ (Ω))N ×N kDϕk2(L2 (Ω))N + S2
kϕk2L∞ (Ω) +
2
α
α
Γ(k)



2k 2


kϕk2L2 (Ω;dµ)
+



 α
∀k > 0, ∀ϕ ∈ H01 (Ω) ∩ L∞ (Ω);
• in Remark 5.6 estimate (5.18) has to be replaced by
(8.13)
Z

1

2

kϕDuk(L2 (Ω))N +
u2 ϕ2 dµ ≤


α

Ω

2


CS2 khkLr (Ω)
 32k 2
2
2
≤ 2 kAk(L∞ (Ω))N ×N kDϕk(L2 (Ω))N + 2 2
kϕk2L∞ (Ω) +
2
α
α Γ(k)

2

k

2


+ kϕkL2 (Ω;dµ)



∀kα> 0, ∀ϕ ∈ H 1 (Ω) ∩ L∞ (Ω);
0
54
D. GIACHETTI, P.J. MARTÍNEZ-APARICIO, AND F. MURAT
• in Remark 5.7 estimate (5.19) has to be replaced by
(8.14)
Z
Z

3
3
2
2 2
2


u φ dµ = kD(φu)k(L2 (Ω))N +
u2 φ2 dµ ≤
kφukH01 (Ω) + α

α Ω

Ω



2
2
khk2Lr (Ω)


≤ 3 32k kAk2(L∞ (Ω))N ×N kDφk2(L2 (Ω))N + 2 CS
kφk2L∞ (Ω) +
α2
α2 Γ(k)2
!


2 khk2 r
2

C
k
L (Ω)

S
2
2
2
2
2


 + α kφkL2 (Ω;dµ) + k kDφkL2 (Ω) + CP (Ω) α2 Γ(k)2 kDφk(L∞ (Ω))N




∀k > 0, ∀φ ∈ D(Ω);
• in Proposition 5.9, estimate (5.40) has to be replaced by
Z


∀δ > 0,
F (x, u)v ≤



{u≤δ}
!


 3 Z X
XZ
ˆ
gˆi Dϕ̂i + f δ +
Zδ (u)gˆi Du ϕ̂i +
≤
(8.15)
2

Ω i∈I
Ω

i∈I

Z




+2δ v dµ;
Ω
• in Proposition 5.12 estimate (5.60) has to be replaced by
Z
2
(8.16) αkDβ(Tk (u))k(L2 (Ω))N + u Γ(Tk (u)) dµ ≤ khkL1 (Ω) ∀k > 0.
Ω
Of course many other small adaptations have to be made here or
there, in particular in the proofs. For example let us mention that in
the second step of the proof of the equivalence result of Proposition B.1
below one uses an approximation of the test function ϕ ∈ H01 (Ω) ∩
∩ L2 (Ω; dµ), first by functions ϕm = Tm (ϕ) ∈ H01 (Ω) ∩ L∞ (Ω), and
then by functions vn = inf(φ+
n , ϕm ) where φn ∈ D(Ω) tends to ϕm in
H01 (Ω) strongly.
8.3. A counterexample to the strong maximum principle in
the case with a zeroth-order term µu with µ ∈ H −1 (Ω) ∩ M+
b (Ω)
In this subsection we give an explicit counterexample which shows
that the strong maximum principle does not in general hold true for the
solutions to (8.4) (or, in order to be mathematically correct, to (8.7))
when the operator involves a zeroth-order term µu with µ ∈ H −1 (Ω) ∩
∩ M+
b (Ω) when N ≥ 3. This counterexample was communicated to us
by Gianni Dal Maso, to whom we express our warmest thanks.
SEMILINEAR PROBLEMS WITH A STRONG SINGULARITY AT u = 0
55
Let Ω be the ball
Ω = {x ∈ RN : |x| < 1},
(8.17)
N ≥ 1,
and let q be the (radial) function defined by
2N
(8.18)
q(|x|) = 2 ∀x ∈ Ω.
|x|
Consider the problem
(
−∆u + q(|x|)u = 0 in Ω,
(8.19)
u=1
on ∂Ω,
or in a mathematically correct sense its weak formulation (compare
with (8.7))
(8.20)

1
2
1
u
Z ∈ H (Ω) ∩ZL (Ω; q(|x|)dx), u − 1 ∈ H0 (Ω),

q(|x|)uvdx = 0 ∀v ∈ H01 (Ω) ∩ L2 (Ω; q(|x|)dx).
DuDv +
Ω
Ω
It is easy to check that for N ≥ 1 the function u defined by
u(x) = |x|2
(8.21)
∀x ∈ Ω
is a solution to (8.20) which satisfies u ≥ 0 in Ω.
Since u(0) = 0, and since u does not coincide with 0 in Ω, the strong
maximum principle does not hold true for problem (8.19) when N ≥ 1
(compare with the statement (C.13) of Appendix C below).
Observe moreover that when N ≥ 3, the measure µ defined by
(8.22)
µ = q(|x|)dx
satisfies
∗ 0
µ ∈ L(2 ) (Ω)
(8.23)
since (2∗ )0 =
2N
N +2
and since
2N
Z 1
2N N +2N −1
ρ
dρ < +∞ when N ≥ 3;
ρ2
0
therefore one has
(8.24)
µ ∈ H −1 (Ω) ∩ M+
b (Ω).
Note that (8.23) does not hold true when N = 1 and N = 2.
Therefore the strong maximum principle does not hold true in general for a solution of (8.7) when µ ∈ H −1 (Ω) ∩ M+
b (Ω) when N ≥ 3.
Actually, the situation described in the explicit counterexample given
by (8.18), (8.20) and (8.21) is not an isolated case. Indeed the results
proved by G. Dal Maso and U. Mosco in [4] and [5] show that when
56
D. GIACHETTI, P.J. MARTÍNEZ-APARICIO, AND F. MURAT
Ω ⊂ RN is any open set with 0 ∈ Ω, and when q : R+ → R+ is any
radial function such that
q ∈ L1loc (]0, +∞[), q ≥ 0,
any local weak solution u to problem (8.19), i.e. any u such that

1
2
u ∈ Hloc
(Ω) ∩


Z Lloc (Ω; q(|x|)dx),
Z
DuDv + uv q(|x|)dx = 0
(8.25)

Ω

 Ω
∀v ∈ H 1 (Ω) ∩ L2 (Ω; q(|x|)dx), supp v ⊂⊂ Ω,
is continuous in Ω (and in particular at 0) (note that there exist nontrivial local weak solutions to problem (8.19) with u ≥ 0, see Theorems 2.4
and 2.10 of [4]). The same authors proved (see Theorem 5.1 of [5]) that
when N ≥ 3 and when
Z
ρ q(ρ)dρ = +∞,
0
then u(0) = 0. This provides a large class of counterexamples to the
strong maximum principle for problems of the type (8.4) with a radial
singularity. In particular the strong maximum principle does not hold
C
true for the radial singularities
, C > 0, λ ≥ 2, when N ≥ 3.
|x|λ
In contrast, it was recently proved by L. Orsina and A. Ponce in [13],
where the above counterexample (8.18), (8.20) and (8.21) is also presented, that the strong maximum principle holds true for the operator
−∆u + V u, with V ≥ 0 non necessarily radial, when V ∈ Lp (Ω) with
p > N2 (see comments after Theorem 1 of [13]). Therefore the strong
C
maximum principle holds true for the radial singularities
, C > 0,
|x|λ
λ < 2.
Appendix A. An useful lemma
In this Appendix we state and prove the following lemma which is
used in some of the proofs of the present paper.
Lemma A.1. If z ∈ H 1 (Ω) and if there exists z and z ∈ H01 (Ω) such
that
z ≤ z ≤ z a.e. in Ω,
then z ∈ H01 (Ω).
SEMILINEAR PROBLEMS WITH A STRONG SINGULARITY AT u = 0
57
Remark A.2. Lemma A.1 is straightforward when ∂Ω is sufficiently
smooth so that the traces of the functions of H 1 (Ω) are defined. Note
that we did not assume any smoothness of ∂Ω.
Proof of Lemma A.1.
Since 0 ≤ z − z ≤ z − z, it is sufficient to consider the case where
z = 0, namely the case where
z ∈ H 1 (Ω) with 0 ≤ z ≤ z, where z ∈ H01 (Ω).
Since z ∈ H01 (Ω), there exists a sequence φn ∈ D(Ω), such that
φn → z
in H01 (Ω).
The function zn defined by
+
+
zn = φ+
n − (φn − z)
belongs to H 1 (Ω) and has compact support (included in the support
of φn ). Therefore
zn ∈ H01 (Ω),
and
zn → z + − (z + − z)+ = z
in H 1 (Ω) as n → +∞.
This implies that z ∈ H01 (Ω).
Appendix B. Three remarks about Definition 3.6
B.1. Equivalence of two definitions in the case of a mild singularity
We prove in this subsection that the Definition 3.6 of a solution to
problem (2.1) given in the present paper coincides with the Definition 3.1 of a solution given in our paper [8] when the singularity is mild
(see (1.5)). In this case the function F satisfies

1

0 ≤ F (x, s) ≤ h(x) + 1 a.e. x ∈ Ω, ∀s > 0,
(B.1)
s
with h(x) ≥ 0 a.e. x ∈ Ω, h ∈ Lr (Ω), r as in (2.5),
1
1
≤ γ + (1 − γ) for s > 0
γ
s
s
and 0 < γ ≤ 1, which results from Young’s inequality. Note that (B.1)
s
is nothing but the case where Γ(s) =
in (2.5 iii).
1+s
Recall that Definition 3.1 in [8] (which is concerned with functions
F which satisfy (1.5) or (B.1)) reads as follows:
as it is easily seen in view of the inequality
58
D. GIACHETTI, P.J. MARTÍNEZ-APARICIO, AND F. MURAT
Definition 3.1 of [8]. Assume that the matrix A and the function F
satisfy (2.3), (2.4) and (B.1). We say that u is a solution to problem
(2.1) in the sense of Definition 3.1 of [8] if u satisfies
(
u ∈ H01 (Ω),
(B.2)
u ≥ 0 a.e. x ∈ Ω,
(B.3)


∀ϕZ∈ H01 (Ω), ϕ ≥ 0, one has




i)
F (x, u)ϕ < +∞,
ZΩ
Z




A(x)DuDϕ =
F (x, u)ϕ.
ii)
Ω
Ω
Proposition B.1. Assume that the matrix A and the function F satisfy (2.3), (2.4) and (B.1). Then u is a solution to problem (2.1) in the
sense of Definition 3.1 of [8] if and only if u is a solution to problem
(2.1) in the sense of Definition 3.6 of the present paper.
Proof.
First step. When u is a solution to problem (2.1) in the sense of
Definition 3.1 of [8], i.e. when u satisfies (B.2) and (B.3), it is clear
that u satisfies (3.10).
On the other hand, let v ∈ V(Ω), v ≥ 0. Then v in particular belongs
to H01 (Ω) and (3.11 i) follows from (B.3 i).
Finally using (3.5) with y = Tk (u), which belongs to H01 (Ω)∩L∞ (Ω),
one has
(B.4)
(
hh−div tA(x)Dv, Tk (u)iiΩ = h−div tA(x)Dv, Tk (u)iH −1 (Ω),H01 (Ω)
∀v ∈ V(Ω),
which implies that (3.11 ii) immediately follows from (B.3 ii).
Second step. Conversely, assume that u is a solution to problem (2.1)
in the sense of Definition 3.6 above.
s
Inequality (B.1) is nothing but (2.5) with Γ(s) =
, so that
1+s
p
1
by (5.58) one has β 0 (s) =
Γ0 (s) =
and β(s) = log(1 + s).
1+s
Proposition 5.12 then implies that β(Tk (u)) = log(1 + Tk (u)) ∈ H01 (Ω)
DTk (u)
and that β 0 (Tk (u)) DTk (u) =
∈ (L2 (Ω))N . This implies that
1 + Tk (u)
DTk (u) ∈ (L2 (Ω))N and that Tk (u) ∈ H01 (Ω) by using Lemma A.1,
SEMILINEAR PROBLEMS WITH A STRONG SINGULARITY AT u = 0
59
since 0 ≤ Tk (u) ≤ (1 + k) log(1 + Tk (u)). Since Gk (u) ∈ H01 (Ω) by
(3.10 iii), this finally implies that u ∈ H01 (Ω). This proves (B.2).
Let us now prove (B.3). Let ϕ ∈ H01 (Ω), ϕ ≥ 0, and let φn be a
sequence such that
φn ∈ D(Ω), φn → ϕ in H01 (Ω) and a.e. in Ω.
For every n, the function vn defined by
vn = inf(φ+
n , ϕ)
belongs to H01 (Ω) ∩ L∞ (Ω), has a compact support which is contained
in Ω and is nonnegative. Therefore (see Remark 3.5 iii)), the function
vn belongs to V(Ω) and can be used as test function in (3.11 ii), giving
(B.5)

t
t
h−div
Z A(x)Dvn , Gk (u)iH −1 (Ω),H01 (Ω) + hh−div A(x)Dvn , Tk (u)iiΩ =
=
F (x, u)vn .
Ω
Since Tk (u) ∈ H01 (Ω)∩L∞ (Ω), using (3.5) with v = vn and y = Tk (u)
gives (B.4) with v = vn , and (B.5) implies that
Z
t
F (x, u)vn .
(B.6)
h−div A(x)Dvn , uiH −1 (Ω),H01 (Ω) =
Ω
Passing to the limit in (B.6) and using Fatou’s Lemma in the right-hand
side gives
Z
t
h−div A(x)Dϕ, uiH −1 (Ω),H01 (Ω) ≥
F (x, u)ϕ,
Ω
which proves that F (x, u)ϕ ∈ L1 (Ω), i.e. (B.3 i); passing again to
the limit in (B.6) and using now Lebesgue’s dominated convergence
Theorem (since vn ≤ ϕ) gives
Z
Z
A(x)DuDϕ =
F (x, u)ϕ,
Ω
i.e. (B.3 ii).
Proposition B.1 is proved.
Ω
B.2. Equivalence of two variants of Definition 3.6 using the requirement “for every k” and the requirement “for a given k0 ”
Definition 3.6 consists in the two assertions (3.10) and (3.11), that in
the present Subsection we will denote by (3.10∀k ) and (3.11∀k ) in order
to emphasize that they have to hold for every k > 0.
60
D. GIACHETTI, P.J. MARTÍNEZ-APARICIO, AND F. MURAT
Let us first observe that Definition 3.6, i.e. the fact that u satisfies
(3.10∀k ) and (3.11∀k ), is equivalent to the fact that u satisfies (3.10bis∀k )
and (3.11∀k ), where (3.10bis∀k ) is

1
(Ω),
i) u ∈ L2 (Ω) ∩ Hloc





ii) u(x) ≥ 0 a.e. x ∈ Ω,
(3.10bis∀k )
iii) Gk (u) ∈ H01 (Ω) ∀k > 0,



iv) ϕTk (u) ∈ H01 (Ω) ∀k > 0, ∀ϕ ∈ H01 (Ω) ∩ L∞ (Ω),



v) β(Tk (u)) ∈ H01 (Ω) ∀k > 0,
where Zthe function β is defined from the function Γ by
sp
β(s) =
Γ0 (t)dt (see (5.58)). Indeed, in view of Proposition 5.12,
0
every u which satisfies (3.10∀k ) and (3.11∀k ) also satisfies (5.59), and
therefore satisfies (3.10bis∀k ); the converse is straightforward.
Actually, we have the following equivalence, which roughly speaking
asserts that in Definition 3.6 “for every k > 0” can be replaced by “for
a given k0 > 0” where k0 can be arbitrarily chosen:
Proposition B.2. Assume that the matrix A and the function F satisfy (2.3), (2.4) and (2.5). Then u is a solution to problem (2.1) in the
sense of Definition 3.6 if and only if for a given k0 > 0 (which can be
arbitrarily chosen) one has
(3.10bisk0 )

1
i) u ∈ L2 (Ω) ∩ Hloc
(Ω),





ii) u(x) ≥ 0 a.e. x ∈ Ω,
iii) Gk0 (u) ∈ H01 (Ω),



iv) ϕTk0 (u) ∈ H01 (Ω) ∀ϕ ∈ H01 (Ω) ∩ L∞ (Ω),



v) β(Tk0 (u)) ∈ H01 (Ω),
SEMILINEAR PROBLEMS WITH A STRONG SINGULARITY AT u = 0
61
(3.11
 k0 )
∀v ∈ V(Ω), v ≥ 0,


X


with − div tA(x)Dv =
ϕ̂i (−div ĝi ) + fˆ in D0 (Ω),




i∈I



1
∞

where
ϕ̂
∈
H
(Ω)
∩
L
(Ω), gˆi ∈ (L2 (Ω))N , fˆ ∈ L1 (Ω),

i
0




one has

 Z
i)
F (x, u)v < +∞,


ZΩ
Z

XZ


t

ii)
A(x)DvDGk0 (u) +
gˆi D(ϕ̂i Tk0 (u)) + fˆTk0 (u) =



Ω
Ω
Ω

i∈I



t
t

= h−div
A(x)Dv, Gk0 (u)iH −1 (Ω),H01 (Ω) + hh−div A(x)Dv, Tk0 (u)iiΩ =

Z




=
F (x, u)v.
Ω
Proof. To prove this equivalence we only have to prove that if u satisfies
(3.10bisk0 ) and (3.11k0 ) for a given k0 > 0, then u satisfies (3.10bis∀k )
and (3.11∀k ), or in other terms u is a solution to problem (2.1) in the
sense of Definition 3.6; the converse is straightforward.
First step. Let us first consider some u which satisfies (3.10bisk0 ) for
a given k0 > 0. Then u satisfies in particular
(B.7)

2
N

χ{u≥k0 } Du ∈ (L (Ω)) ,
χ{u≤k0 } ϕDu ∈ (L2 (Ω))N ∀ϕ ∈ H01 (Ω) ∩ L∞ (Ω),

χ
β 0 (u)Du ∈ (L2 (Ω))N .
{u≤k0 }
p
Since β 0 (s) =
Γ0 (s) for every s > 0, the function β 0 , like the
0
function Γ (see (2.5 ii)), is continuous and satisfies β 0 (s) > 0 for every
s > 0. Therefore for every a and b with 0 < a < b < +∞ one has
0 < min β 0 (t) ≤ max β 0 (t) < +∞;
a≤t≤b
a≤t≤b
together with (B.7), this implies that for every k > 0

2
N

χ{u≥k} Du ∈ (L (Ω)) ,
χ{u≤k} ϕDu ∈ (L2 (Ω))N ∀ϕ ∈ H01 (Ω) ∩ L∞ (Ω),

χ
β 0 (u)Du ∈ (L2 (Ω))N .
{u≤k}
62
D. GIACHETTI, P.J. MARTÍNEZ-APARICIO, AND F. MURAT
On the other hand, since β is nondecreasing, one has, when k > k0 ,


0 ≤ Gk (s) ≤ Gk0 (s) ∀s > 0,

0 ≤ Tk (s) = Tk0 (s) + Tk−k0 (Gk0 (s)) ∀s > 0,

β(k)

0 ≤ β(Tk (s)) ≤
β(Tk0 (s)) ∀s > 0,
β(k0 )
and when k < k0 ,

k0 − k


0 ≤ Gk (s) ≤ β(k) β(Tk0 (s)) + Gk0 (s) ∀s > 0,
0 ≤ Tk (s) ≤ Tk0 (s) ∀s > 0,



0 ≤ β(Tk (s)) ≤ β(Tk0 (s)) ∀s > 0.
Using Lemma A.1, this implies that every u which satisfies (3.10bisk0 )
also satisfies (3.10bis∀k ).
Second step. Let us now consider some u which satisfies (3.10bisk0 )
and (3.11k0 ).
As in (5.5), we define for every a and b with 0 < a < b < +∞, the
function Sa,b as
Sa,b (s) = Tb (s) − Ta (s) ∀s ≥ 0,
and we observe that since Gk (s) + Tk (s) = s, one has
Ga (s) − Gb (s) = Tb (s) − Ta (s) ∀s ≥ 0.
(B.8)
Since we have proved that (3.10bisk0 ) implies (3.10bis∀k ), the function Sa,b (u) belongs to H01 (Ω) ∩ L∞ (Ω) for every 0 < a < b < +∞, and
therefore, for every v ∈ V(Ω) with

X
−div tA(x)Dv =
ϕ̂i (−div ĝi ) + fˆ in D(Ω),
i∈I

where ϕ̂i ∈ H01 (Ω), ĝi ∈ (L2 (Ω))N , fˆ ∈ L1 (Ω),
one has (see (3.5))
Z
Z
XZ
t
A(x)DvDSa,b (u) =
ĝi D(ϕ̂i Sa,b (u)) + fˆSa,b (u),
Ω
i∈I
Ω
Ω
or in other terms, using (B.8),
Z
t


 A(x)Dv(DGa (u) − DGb (u)) =
Ω
Z
XZ

ĝi D(ϕ̂i (Tb (u) − Ta (u))) + fˆ(Tb (u) − Ta (u)).

=
i∈I
Ω
Ω
SEMILINEAR PROBLEMS WITH A STRONG SINGULARITY AT u = 0
63
Since Ga (u), Gb (u), ϕ̂i Ta (u) and ϕ̂i Tb (u) belong to H01 (Ω), while Ta (u)
and Tb (u) belong to L∞ (Ω), we can split the differences and we obtain
Z
Z
XZ

t

ĝi D(ϕ̂i Ta (u)) + fˆTa (u) =

 Ω A(x)DvDGa (u) +
Ω
i∈I ΩZ
Z
Z
X

t

A(x)DvDGb (u) +
ĝi D(ϕ̂i Tb (u)) + fˆTb (u),

=
Ω
i∈I
Ω
Ω
for every a, b with 0 < a < b, and therefore for every a > 0 and b > 0.
Taking a = k and b = k0 implies that (3.11∀k ii) holds true whenever
(3.10k0 ) and (3.11k0 ii) hold true.
The desired result is proved.
Remark B.3. If one knows in advance that a function u satisfies(2)
u ∈ L∞ (Ω),
(B.9)
then Definition 3.6 becomes simpler. Indeed in this case, taking
k0 ≥ kukL∞ (Ω) in Proposition B.2, one has Gk0 (u) = 0 and Tk0 (u) = u,
so that Definition 3.6 is equivalent to the two assertions

1
i) u ∈ L∞ (Ω) ∩ Hloc
(Ω),



ii) u(x) ≥ 0 a.e. x ∈ Ω,
(B.10)

iii) ϕu ∈ H01 (Ω) ∀ϕ ∈ H01 (Ω) ∩ L∞ (Ω),



iv) β(u) ∈ H01 (Ω),


∀v ∈ V(Ω), v ≥ 0,


X

t


with
−
div
A(x)Dv
=
ϕ̂i (−div ĝi ) + fˆ in D0 (Ω),




i∈I


∞
1

where ϕ̂i ∈ H0 (Ω) ∩ L (Ω), gˆi ∈ (L2 (Ω))N , fˆ ∈ L1 (Ω),




one has
Z
(B.11)

i)
F (x, u)v < +∞,



Ω
Z

XZ




ii)
gˆi D(ϕ̂i u) + fˆu = hh−div tA(x)Dv, uiiΩ =



Ω
Ω

 Zi∈I



F (x, u)v.
=
Ω
(2)this
is the case (see [9, Section 5]) if in place of assumption (2.5 i), the function
h is assumed to satisfy
N
h ∈ Lt (Ω), t >
if N ≥ 2, t = 1 if N = 1
2
64
D. GIACHETTI, P.J. MARTÍNEZ-APARICIO, AND F. MURAT
B.3. A variant of Definition 3.6 using another space W(Ω) of
test functions
Definition 3.6 above makes use in (3.11) of test functions v which
belong to the space V(Ω). Actually, as far as the results of the present
paper are concerned, another definition of the solution could be used,
where in (3.11) the space V(Ω) of test functions would be replaced by
the space W(Ω) of test functions w of the form
X
w=
ϕi ψi
i∈I
where I is finite and where ϕi and ψi belong to H01 (Ω) ∩ L∞ (Ω).
This space W(Ω) is a vector space, which in view of Remark 3.5 i) is
a subspace of V(Ω). Moreover since 2ϕi ψi = (ϕi + ψi )2 − (ϕi − ψi )2 , the
space W(Ω) is generated by the functions w = ϕ2 , with ϕ ∈ H01 (Ω) ∩
∩L∞ (Ω), which are very simple to use.
It can be seen that all the results obtained in the present paper
(namely the existence of a solution, its stability with respect to the
variation of the right-hand side, the a priori estimates obtained in Section 5, as well as the uniqueness of the solution when F (x, s) is nonincreasing in s) can be obtained with this (new) definition using the
space W(Ω) in place of V(Ω).
We have nevertheless chosen to present the results of the present
paper in the framework of the space V(Ω), because it seems to us
that the use of the smaller space W(Ω) would not allow us to treat
homogenization problems with many small holes, a problem that we
study in our paper [10].
Of course a solution defined by using the space V(Ω) is also a solution
defined by using the smaller space W(Ω). The converse is unclear to us,
unless we assume that F (x, s) is nonincreasing in s. Indeed in this case
the uniqueness of the two types of solutions and their approximation
by problems for Fn (x, s) = Tn (F (x, s)) easily allow one to prove that
they coincide.
Appendix C. Study of the set
where the solution u is equal to zero
Every solution u to problem (2.1) in the sense of Definition 3.6 is
a nonnegative function, which could in principle vanish on a set of
(strictly) positive measure. We will prove in this appendix that this is
not the case.
SEMILINEAR PROBLEMS WITH A STRONG SINGULARITY AT u = 0
65
A first observation in this direction is the following: the nonnegative
measurable function F (x, u(x)) can take infinite values when u(x) = 0.
For every function v which is measurable and nonnegative, the integral
Z
F (x, u(x))v is then correctly defined as a number which belongs to
Ω
[0, +∞]. But assumption (3.11 i) on the solution u requires that this
number is finite for every v ∈ V(Ω), v ≥ 0, which implies (see (3.14))
that F (x, u(x)) ∈ L1loc (Ω). Therefore, when u is a solution to problem
(2.1) in the sense of Definition 3.6, we have
F (x, u(x)) is finite a.e. x ∈ Ω,
(C.1)
which implies that
(C.2)
meas{x ∈ Ω : u(x) = 0 and F (x, 0) = +∞} = 0,
or equivalently that
(
{x ∈ Ω : u(x) = 0} ⊂ {x ∈ Ω : F (x, 0) < +∞}
(C.3)
except on a set of zero measure.
A result stronger than (C.2) (see (C.6)) is given in the following
Proposition C.1, and an even stronger result will be given in Proposition C.3 below (note however that the latest one uses the strong
maximum principle).
Proposition C.1. Assume that the matrix A and the function F satisfy (2.3), (2.4) and (2.5). Then every solution u to problem (2.1) in
the sense of Definition 3.6 satisfies
(C.4)
meas{x ∈ Ω : u(x) = 0 and 0 < F (x, 0) ≤ +∞} = 0,
and
Z
F (x, u)v = 0 ∀v ∈ V(Ω), v ≥ 0.
(C.5)
{u=0}
Remark C.2. Assertion (C.4) is equivalent to
(
{x ∈ Ω : u(x) = 0} ⊂ {x ∈ Ω : F (x, 0) = 0}
(C.6)
except for a set of zero measure,
(compare with (C.3)) and also equivalent to
(
{x ∈ Ω : 0 < F (x, 0) ≤ +∞} ⊂ {x ∈ Ω : u(x) > 0}
(C.7)
except for a set of zero measure.
66
D. GIACHETTI, P.J. MARTÍNEZ-APARICIO, AND F. MURAT
On the other hand, assertion (C.5) is equivalent to
Z
Z
F (x, u)v =
F (x, u)v ∀v ∈ V(Ω), v ≥ 0.
(C.8)
Ω
{u>0}
Proof of Proposition C.1.
Proposition 5.11 asserts that every u solution to problem (2.1) in the
sense of Definition 3.6 satisfies (5.42), i.e. (C.5).
Writing

{u = 0} =


 = {u = 0} ∩ {F (x, 0) = 0} ∪
(C.9)


∪ {u = 0} ∩ {0 < F (x, 0) ≤ +∞}
therefore implies that (C.5) is equivalent to
Z
(C.10)
F (x, u)v = 0,
∀v ∈ V(Ω), v ≥ 0.
{u=0}∩{0<F (x,0)≤+∞}
Since every φ ∈ D(Ω) belongs to V(Ω) (see Remark 3.5 iv) this last
assertion is equivalent to (C.4).
The following Proposition asserts that every solution u to problem
(2.1) in the sense of Definition 3.6 actually satisfies u(x) > 0 almost
everywhere in Ω, except in the case where u = 0 in the whole of Ω. This
is one of the keypoints of the paper [2] by L. Boccardo and L. Orsina,
which was the source of inspiration of the present paper.
This assertion is much stronger than (C.6), but its proof uses the
strong maximum principle. Note however that there are situations
different from (but close to) the present one where the strong maximum
principle does not hold true, see in particular the counterexample given
by (8.18), (8.20) and (8.21) in Subsection 8.3 above. This is the reason
why we stated the weaker result (C.6).
Note finally that Proposition C.3 and Remark C.4 below are the
only points of the present paper where the strong maximum principle
is used.
Proposition C.3. Assume that the matrix A and the function F satisfy (2.3), (2.4) and (2.5). Then every solution u to problem (2.1) in
the sense of Definition 3.6 satisfies
(C.11)
either
meas{x ∈ Ω : u(x) = 0} = 0
or u = 0 in Ω.
SEMILINEAR PROBLEMS WITH A STRONG SINGULARITY AT u = 0
67
Proof.
First step. In this step we recall the statement of the strong maximum principle as it can be found in the book [11] by D. Gilbarg and
N. Trudinger, or more exactly the variant of Theorem 8.19 of [11] where
u is replaced by −u. In this variant, Theorem 8.19 of [11] reads as

1

Let u ∈ H (Ω) which satisfies Lu ≤ 0.
(C.12)
If for some ball B ⊂⊂ Ω one has inf B u = inf Ω u ≤ 0,

the function u is constant in Ω.
In the notation (8.1) and (8.2) of [11], one has Lu = div A(x)Du and
Lu ≤ 0 is nothing but −div A(x)Du ≥ 0. Therefore (C.12) implies
that for any open bounded set ω ⊂ RN one has
(C.13)

1
0

Let u ∈ H (ω) with − div A(x)Du ≥ 0 in D (ω).
If u ≥ 0 a.e. in ω and if inf B0 u = 0 for some ball B0 ⊂⊂ ω,

then u = 0 in ω,
where we have used that the fact that when u is a constant in ω with
inf B0 u = 0, then u = 0 in ω.
Second step. Consider now u which is a solution to problem (2.1) in
the sense of Definition 3.6. Then by Proposition 3.8, the function u
satisfies
1
(C.14) u ∈ Hloc
(Ω),
−div A(x)Du ≥ 0 in D0 (Ω),
u ≥ 0 a.e. in Ω.
Since u ≥ 0 almost everywhere in Ω we have the alternative:
(
either inf B u > 0 for every ball B ⊂⊂ Ω,
or there exists a ball B0 ⊂⊂ Ω such that inf B0 u = 0.
In the first case we have meas{x ∈ Ω : u(x) = 0} = 0. In the second
1
case, since u belongs only to Hloc
(Ω), we consider any open set ω such
that B0 ⊂⊂ ω ⊂⊂ Ω. Then u = 0 in ω for any ω, and therefore u = 0
in Ω. This proves (C.11).
Remark C.4. If u = 0 is a solution to problem (2.1) in the sense of
Definition 3.6, then Proposition 3.8 implies that F (x, 0) ≡ 0.
Conversely, if F (x, 0) 6≡ 0, u = 0 is not a solution to problem (2.1)
in the sense of Definition 3.6 and Proposition C.3 then implies that
(C.15)
u(x) > 0 a.e. x ∈ Ω.
68
D. GIACHETTI, P.J. MARTÍNEZ-APARICIO, AND F. MURAT
Acknowledgments. The authors would like to warmly thank
Gianni Dal Maso and Luc Tartar for their friendly help, and specially
Gianni Dal Maso for authorizing them to reproduce his Counterexample presented in Subsection 8.3. They also would like to thank
Lucio Boccardo, Juan Casado-Dı́az and Luigi Orsina whose papers [1]
and [2] inspired the present work. They finally would like to thank
their own institutions (Dipartimento di Scienze di Base e Applicate per
l’Ingegneria della Facoltà di Ingegneria Civile e Industriale di Sapienza
Università di Roma, Departamento de Matemática Aplicada y Estadı́stica de la Universidad Politécnica de Cartagena, Laboratoire
Jacques-Louis Lions de l’Université Pierre et Marie Curie Paris VI et
du CNRS) for providing the support of reciprocal visits which allowed
them to perform the present work. The work of Pedro J. Martı́nezAparicio has been partially supported by the grant MTM2015-68210-P
of the Spanish Ministerio de Economı́a y Competitividad (MINECOFEDER), the FQM-116 grant of the Junta de Andalucı́a and the grant
Programa de Apoyo a la Investigación de la Fundación Séneca-Agencia
de Ciencia y Tecnologı́a de la Región de Murcia 19461/PI/14.
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Daniela Giachetti
Dipartimento di Scienze di Base e Applicate per l’Ingegneria
Facoltà di ingegneria civile e industriale
Sapienza Università di Roma
Via Scarpa 16, 00161 Roma, Italy
E-mail address: [email protected]
Pedro J. Martı́nez-Aparicio
Departamento de Matemática Aplicada y Estadı́stica
Universidad Politécnica de Cartagena
Paseo Alfonso XIII 52, 30202 Cartagena (Murcia), Spain
E-mail address: [email protected]
François Murat
Laboratoire Jacques-Louis Lions et CNRS
Université Pierre et Marie Curie
Boı̂te Courrier 187, 75252 Paris Cedex 05, France
E-mail address: [email protected]