Generalized logistic equation with indefinite
weight driven by the square root of the Laplacian
Alessio Marinelli
Department of Mathematics
University of Trento
Via Sommarive 14, 38123 Povo (TN) - Italy
[email protected]
Dimitri Mugnai
Department of Mathematics and Computer Sciences
University of Perugia
Via Vanvitelli 1, 06123 Perugia - Italy
[email protected]
Abstract
We consider an elliptic problem driven by the square root of the Laplacian in presence of a general logistic function having an indefinite weight.
We prove a bifurcation result for the associated Dirichlet problem via regularity estimates of independent interest when the weight belongs only to
certain Lebesgue spaces.
Keywords: Indefinite potential, principal eigenfunction, nonlinear regularity,
existence and multiplicity theorems, nonlinear maximum principle
2000AMS Subject Classification: 35J20, 35J65, 58E05
1
Introduction
One of the most popular equations in Biology, and in particular in population
dynamics, is the logistic equation, which we write as
cut (x, t) = ∆u(x, t) + λ(β(x)u(x, t) − u2 (x, t)) x ∈ Ω ⊂ RN , t > 0.
(1)
Here Ω is a domain in RN where the population having nonnegative density u
lives and c, λ are positive constants. However, it is well known that the heat
operator may be too rigid to describe the possible interaction of the specie in the
whole of Ω (for instance, see [5], [6], [19], [21] and the references therein), and for
this fact a nonlocal operator may be more useful than a local one.
Moreover, a first natural step consists in studying steady states for these new
equations (like in [5]). For this reason, in this paper, the diffusion is not decribed
by classical Laplacian
operator (or its generalizations), but by the square root
√
of −Laplacian, −∆. This operator can be defined in several ways, and we
1
Logistic square root of the Laplacian with indefinite weight
2
follow the approach developed in the pioneering works of Caffarelli, Silvestre
[10], Caffarelli, Vasseur [11], Cabré, Tan [9], to which we refer in Section 2 for
the precise mathematical description and properties. Here, we only say that it
is becoming more and more popular for its applications in Physics, Probability,
Finance and Biology. In particular, we recall the pure periodic logistic fractional
equation in RN studied in [7] - where the behaviour of level set of solutions is
investigated - , the evolution Fisher-KKP
equation in RN driven by the generator
√
of a Feller semigroup (for instance, −∆) studied in [8] - where the authors prove
that the stable state invades the unstable one with a front position which grows
exponentially in time - , and the general fractional porous medium equation in
RN studied in [13] - where existence and uniqueness are established.
The problem we are concerned with has homogeneous Dirichlet boundary
conditions, so that the population is confined in a bounded domain Ω ⊂ RN ,
N ≥ 2. Hence, we shall study the following problem:
(√
−∆u(x) = λ(β(x)u(x) − g(x, u(x))) in Ω
(Pλβ )
u(x) = 0
on ∂Ω.
Here the nonlinear term g(x, u) is a self–limiting factor for the population which
generalizes the classical quadratic nonlinearity in (1), and whose properties will
be introduced later on; we only emphasize the fact that, unlike similar variational
problems, we don’t impose any growth condition at infinity for g.
The weight β : Ω −→ R corresponds to the birth rate of the population
and, as main contribution of this paper, we suppose that it is sign changing and
(generally) unbounded, so that both contributions to development and limitation
of the population are possible. To our best knowledge, no results are known for
unbounded sign changing weights, while this situation is clearly possible. Under
this setting, we will establishes a relation between the parameter λ > 0 and
the existence of a nontrivial stationary solution of the problem, showing that
if λ > 0 is large, non trivial steady state are possible, though the birth rate
β is mainly negative in the domain (as it happens in regions with natural or
artificial difficulties which prevent proliferation); on the other hand if λ is small,
the population is not safely protected in regions of disadvantage and steady state
cannot exist (see Theorem 4.1). Let us also remark that λ large corresponds to
a small diffusion and λ small corresponds to a large diffusion in (1).
We recall that logistic–type equations have been extensively studied and, in
addition to the papers cited above, any bibliography would be incomplete. We
only add [20], where steady states of equation (1) where found when Ω = RN and
β is sign–changing. Hence our result, is the counterpart of the results therein in
the case of problem (Pλβ ).
The paper is organized as follows. Section 2 introduces the mathematical
background we shall need in the following sections. In particular we recall some
basic facts on the square root of the Laplacian and some fundamental inequalities.
In Section 3 we will introduce the main character of this paper (i.e. the bifurcation
parameter λ∗ ), and we will prove a general maximum principles (see Proposition
3.2) and a regularity result (see Proposition 3.3) of independent interest. In the
last section we will prove a bifurcation theorem for problem (Pλβ ): via critical
Logistic square root of the Laplacian with indefinite weight
3
point theory and truncation techniques we will show that problem (Pλ ) admits
a nontrivial solution if and only if λ > λ∗ , and, with additional but natural
conditions, we will show that nontrivial solutions bifurcate from λ∗ (see Theorems
4.1 and 4.2).
2
Mathematical Background
In this section we briefly recollect some definitions and inequalities we will use
throughout the paper.
From now on, Ω ⊆ RN will denote a bounded smooth domain, and we denote
by C = Ω × (0, +∞) the half cylinder with base Ω × {0} - which we shall identify
with Ω - , and lateral boundary ∂L C = ∂Ω × [0, +∞).
Then, we introduce the Sobolev Space
1
H0,L
(C) = v ∈ H 1 (C) | v = 0 a.e. on ∂L C ,
equipped with the norm
Z
kvkH 1
0,L (C)
2
21
|Dv(x, y)| dxdy
=
,
C
induced by the scalar product
Z
Dv · Dz dxdy,
1
v, z ∈ H0,L
(C).
C
As usual, a crucial rôle is played by trace operator
1
1
T rΩ : H0,L
(C) −→ H 2 (Ω),
(2)
which is a continuous map (see [9, Lemma 2.6]), and which gives several information, which we recall in the following. In particular, we will repeatedly use
the following
Lemma 2.1 (Sobolev trace inequality, Lemma 2.3, [9]). Let N ≥ 2 and 2] =
2N
1
N −1 . Then there exists a constant C = C(N ), such that, for all v ∈ H0,L (C)
1]
Z
21
2
2
|v(x, 0)| dx
≤C
|Dv(x, y)| dxdy
.
Z
2]
(3)
C
Ω
From Lemma 2.1 it is immediate to prove
Lemma 2.2 (Lemma 2.4, [9]). Let N ≥ 2 and 1 ≤ q ≤ 2] = N2N
−1 . Then there
1
exists a positive C = C(N, q, |Ω|), such that, for all v ∈ H0,L
(C)
Z
q
|v(x, 0)| dx
Ω
q1
Z
≤C
2
|Dv(x, y)| dxdy
C
If N = 1, inequality (4) holds for all q ∈ [1, ∞).
21
.
(4)
Logistic square root of the Laplacian with indefinite weight
4
Moreover, a stronger useful relation, which will be frequently used, is given
by the following compact embedding:
Lemma 2.3 (Lemma 2.5,[9]). Let 1 ≤ q < 2] if N ≥ 2 and q ∈ [1, ∞) if N = 1;
then
1
T rΩ (H0,L
(C)) ,→,→ Lq (Ω).
The trace operator in (2) permits to introduce the trace space on Ω × {0} of
1
functions in H0,L
(C) as
1
1
V0 (Ω) := u = T rΩ v | v ∈ H0,L
(C) ⊂ H 2 (Ω),
equipped with the norm
kukV0 (Ω) =
2
kuk 21
H
Z
+
Ω
u2
d(x)
12
,
where d(x) = dist(x, ∂Ω). However, there exists another characterization of
V0 (Ω):
Lemma 2.4 ( Lemma 2, 7, [9]). The following identity holds:
Z
1
u2
V0 (Ω) = u ∈ H 2 (Ω) |
dx < ∞ .
Ω d(x)
Here, and in the following, we have denoted by x a generic point of Ω and by
y a generic point in (0, ∞).
Next we introduce the following minimizing problem: fix u ∈ V0 (Ω) and
consider
Z
2
1
inf
|Dv| dxdy | v ∈ H0,L (C), v(., 0) = u in Ω .
(5)
C
Proposition 2.1 (Lemma 2.8, [9]). For every u ∈ V0 (Ω) there exists a unique
minimizer v of (5), called the harmonic extension of u to C which vanishes
on ∂L C.
Such a function is denoted by v =h-ext(u).
Remark 2.1. If {ek }k is the orthonormal frame of L2 (Ω) obtained by the eigenfunctions of −∆ with homogeneous Dirichlet boundary conditions and with asso∞
X
ciated eigenvalues {λk }k , given u ∈ V0 (Ω), we can write u =
bk ek (x). Hence,
k=1
1
by Proposition 2.1, we get that v =h-ext(u) ∈ H0,L
(C) is given by
v(x, y) =
∞
X
1
2
bk ek (x)e−λk y .
k=1
Finally, let us define the square root of the Laplacian (or half-Laplacian) in
a bounded domain Ω of RN . For any continuous function u in Ω̄, there exists a
unique function v such that


in C
 ∆C v = 0
(6)
v(x, 0) = u(x) on Ω × {0} ' Ω,


v=0
on ∂L C,
Logistic square root of the Laplacian with indefinite weight
5
where ∆C denotes the Laplacian operator in C, so that v is the generalized harmonic extension of u in C which is 0 on ∂L C.
Now, consider the operator T defined as
T u(x) := −
∂v
(x, 0).
∂y
It is readily seen that the system


 ∆C w = 0
∂v
w(x, 0) = − ∂y
(x, 0) = T u(x)


w=0
(7)
in C
on Ω × {0} ' Ω,
on ∂L C,
∂v
admits the unique solution w(x, y) = − ∂y
(x, y), and thus by (6) we immediately
have
∂w
∂2v
T (T u)(x) = −
(x, 0) =
(x, 0) = (−∆v)(x, 0).
∂y
∂y 2
In conclusion, T 2 = ∆, i.e the operator T mapping the Dirichlet datum u to the
∂v
Neumann datum − ∂y
(·, 0) is a square root of the Laplacian −∆ in Ω.
In light of the previous “Dirichlet to Neumann” procedure, by Remark 2.1 we
have that
∞
X
√
∂v ∂v 1/2
−∆u(x) =
=
−
=
bk λk ek (x),
∂ν Ω×{y=0}
∂y Ω×{y=0}
k=1
where ν = −y. Moreover, it is natural to give the following
1/2
Definition 2.1. We say that u ∈ H0 (Ω) is a weak solution of
(√
−∆u = g(x, u) in Ω
u=0
su ∂Ω,
1
whenever the harmonic extension v ∈ H0,L
(C) of u is a weak solution of


in C
∆v = 0
v=0
on ∂L C

 ∂v
=
g(x,
u)
on
Ω × {0} ,
∂ν
that is if
Z
Z
Dv · Dξ dxdy =
C
g(x, v(x, 0))ξ(x, 0) dx
(*)
(**)
1
∀ξ ∈ H0,L
(C).
Ω×{0}
An important regularity result for
√
−∆ is available:
Theorem 2.1 (Theorem 1.9, [9])). Let Ω be a bounded domain of RN of class
C 2,α , α ∈ (0, 1) . If g ∈ V0 (Ω)∗ and u is a weak solution of
(√
−∆u = g(x) in Ω
(P)
u=0
on ∂Ω,
then:
Logistic square root of the Laplacian with indefinite weight
6
1. If g ∈ L2 (Ω), then u ∈ H01 (Ω).
2. If g ∈ H01 (Ω), then u ∈ H 2 (Ω) ∩ H01 (Ω).
3. If g ∈ L∞ (Ω), then u ∈ C α (Ω).
4. If g ∈ C α (Ω) and g|∂Ω ≡ 0, then u ∈ C 1,α (Ω).
5. If g ∈ C 1,α (Ω) and g|∂Ω ≡ 0, then u ∈ C 2,α (Ω).
Of course, applying Theorem 2.1 to weak solutions of
(√
−∆u = f (x, u) in Ω
u=0
on ∂Ω,
(Q)
we can obtained related results for nonlinear problems.
3
The bifurcation parameter
We consider the problem
(√
−∆u = λ(βu − g(x, u)) in Ω
u=0
on ∂Ω,
(Pλβ )
where N ≥ 2, λ ∈ R, Ω ⊂ RN is a bounded domain of class C 2,α for some
0 < α ≤ 1, g : Ω × R −→ R is a Carathéodory perturbation (i.e., for all s ∈ R
the map x 7→ g(x, s) is measurable and for a.e. x ∈ Ω the map x 7→ g(x, s) is
continuous). Concerning the function β, we assume that it is a sign changing
measurable weight (for a first presentation of such classes of weights, we refer to
[24]).
In view of the considerations of the previous section, studying problem (Pλβ )
is equivalent to studying the extension problem in the cylinder


in C
−∆v = 0
(Qβλ )
v=0
on ∂L (C)

 ∂v
∂ν = λ(βv − g(x, v)) in Ω × {0} .
Of course, (Qβλ ) has a variational structure, so that v solves (Qβλ ) if and only if
1
it is a critical point for the functional I : H0,L
(C) → R defined as
Z
Z
1
2
|Du| dxdy − λ
F (x, u) dx,
(8)
I(u) =
2 C
Ω×{0}
where
Z
u
[βs − g(x, s)]ds.
F (x, u) =
0
From now on, all integrals of the form
Z
Z
will be simply denoted by
.
Ω×{0}
Ω
Logistic square root of the Laplacian with indefinite weight
7
Now, we set
λ∗ = inf
Z
2
1
|Du| dxdy : u ∈ H0,L
(C) with
C
Z
βu2 = 1 .
(9)
Ω
In order to ensure that λ∗ is well defined, we impose the following condition on
the weight function β:
H(β): β ∈ Lq (Ω) with q > N and β + = max{β, 0} =
6 0.
Remark 3.1. The assumption β + = max{β, 0} =
6 0 implies that the set
Z
1
2
u ∈ H0,L (C) with
βu = 1
Ω
is not empty, so that λ∗ is well defined.
Proposition 3.1. If hypothesis H(β) holds, then λ∗ > 0.
R
1
1
Proof. Let u ∈ H0,L
(C) be such that Ω βu2 = 1. Since u ∈ H0,L
(C), then
]
u ∈ L2 (Ω), and we note that
1
+
N
1
N
N −1
=
N −1
1
+
= 1.
N
N
Then, using Hölder’s inequality and Lemma 2.1, we get
Z
q−N 2
2
1=
βu2 dx ≤ β + LN (Ω) kukL2] (Ω) ≤ |Ω| qN β + Lq (Ω) kuk2] (Ω)
Ω
2
≤ C1 β + Lq (Ω) kDukL2 (C)
for some universal constant C1 > 0.
Hence,
Z
2
inf1
|Du| dxdy = λ∗ ≥ (C1 β + Lq (Ω) )−1 > 0.
u∈ H0 (C)
C
We introduce the following general maximum principles.
Proposition 3.2. Let Ω be a smooth bounded domain of RN and let u ∈ C 1 (Ω)
be a solution of
√

 −∆u + B(x, u, Du) ≥ 0 in Ω
u≥0
in Ω


u=0
on ∂Ω,
with B = B(x, s, ξ) of class C 1 in ξ, B ∈ Liploc in s and such that B(x, 0, 0) = 0
for a.e. x ∈ Ω. Then u > 0 in Ω or u ≡ 0 in Ω.
Logistic square root of the Laplacian with indefinite weight
8
Proof. Let v =h-ext(u) be the harmonic extension of u, so that v satisfies

−∆v = 0
in



v ≥ 0
in

v
=
0
on


 ∂v
∂ν ≥ −B(x, v(x, 0), Dv(x, 0)) in
C,
C,
∂L (C),
Ω.
Suppose by contradiction that u is not identically zero and that there exists
x0 ∈ Ω such that u(x0 ) = 0, so that v(x0 , 0) = 0. By [22, Theorem 2.7.1] we find
∂ν v(x0 , 0) < 0.
(10)
By (10), the assumptions on B and the equivalence between problem (P ) and
(P 0 ), we find
√
0 > −∆u(x0 ) ≥ −B(x0 , u(x0 ), Du(x0 )) = 0,
and a contradiction arises.
Remark 3.2. In the case of weak solution, the previous result still holds true
by [22, Theorem 5.4.1].
Next, let us consider the problem
(√
−∆u = λ∗ βu in Ω
u=0
on ∂Ω.
(Pλ∗ )
We start with the following regularity result (see [16] for a related result in
RN ).
Proposition 3.3. Let Ω be a smooth bounded domain of RN and let u be a weak
solution of (Pλ ). If H(β) holds, then u ∈ C α (Ω).
1
Proof. Set v =h-ext(u), so that v ∈ H0,L
(C) solves


−∆v = 0
v=0

 ∂v
∂ν = λβv
in C
on ∂L (C)
in Ω × {0} .
Now, we will use the Moser iteration technique. If M > 0 we define vM =
1
min {v + , M } and thus vM ∈ H0,L
(C) ∩ L∞ (C). For any nonnegative number k
2k+1
2k
we set φ = vM
, so that Dφ = (2k + 1)vM
DvM .
2k+1
By definition of weak solution with φ = vM
as test function, we get
Z
Z
2k+1
2k
(2k + 1) vM
Dv · DvM dxdy = λ
βvvM
dx,
C
that is
Ω
Z
(2k + 1)
C
2k
vM
|DvM |2 dxdy = λ
Z
Ω
2k+2
βvM
dx.
(11)
Logistic square root of the Laplacian with indefinite weight
9
We note that
Z
(2k + 1)
(2k + 1)
k+1
k+1
< DvM
, DvM
>
(k + 1)2
Z
(2k + 1)
k+1 2
|DvM
| dxdy.
=
(k + 1)2 C
2k
vM
|DvM |2 dxdy =
C
(12)
In addition, by applying the Hölder inequality to the right-hand side of (11), we
obtain
Z
2
2k+2
(13)
λ
βvM
dx ≤ λ β + Lq (Ω) (v + )k+1 L2q0 (Ω) .
Ω
By applying (12) and (13), from (11) one gets
Z
2
(2k + 1)
k+1 2
|DvM
| dxdy ≤ λ β + q (v + )k+1 2q0 .
2
(k + 1) C
(14)
Since lim vM (x, y) = v + (x, y) for a.e. (x, y) ∈ C, applying Fatou’s Lemma,
M →∞
we get
1/2 + k+1 k+1 √ 1
(v )
≤√
λ β + q (v + )k+1 2q0 .
(15)
H0,L (C)
2k + 1
By Lemma 2.1, we get the existence of S > 0 such that
√ + k+1 ] ≤ √k + 1 S λ β + 1/2 (v + )k+1 0 ,
(v )
2
2q
q
2k + 1
(16)
where, with some abuse of notation, we have set kvkLq (Ω) = kv(·, 0)kLq (Ω×{0})
1
(C) and all q ≥ 1.
for any v ∈ H0,L
Since q > N , then 2q 0 < 2] . Now, define k0 = 0 and by recurrence
2(kn + 1)q 0 = (kn−1 + 1)2] .
It is easy see that ki > ki−1 and that kn → ∞ as n → ∞. Moreover, (16)
implies that
0
]
0
if (v + ) ∈ L2(ki−1 +1)q (Ω) then (v + ) ∈ L(ki−1 +1)2 (Ω) = L2(ki +1)q (Ω),
and hence (v + ) ∈ Lr (Ω) for every r ≥ 1. Moreover, by (15) we obtain that
Z
|D(v + )ki +1 |2 ≤ Ci for every i ≥ 0,
C
+ ki +1
1
or equivalently (v )
∈ H0,L
(C) for every i ≥ 0.
Analogously, choosing vM := min{M, v − }, we can prove that v − ∈ Lr (Ω) for
every r ≥ 1.
Now, since v ∈ Lr (Ω) for all r < ∞, then βv ∈ Lq (Ω) with q > q > N .
Now, following the proofRof Proposition 3.1 (iii) in [9], for (x, y) ∈ C we introduce
y
the function w(x, y) = 0 v(x, t)dt, which turns out to solve the homogeneous
Dirichlet problem
(
−∆w(x, y) = λβ(x)v(x, 0) in C,
w=0
on ∂C.
Logistic square root of the Laplacian with indefinite weight
10
Then, performing the odd reflection wodd of w in Ω×R, by the Calderón-Zygmund
inequality, we obtain that wodd ∈ W 2,q (Ω × (−R, R)) for every R > 0. In
particular, w ∈ C 1,α (Ω̄), and so, by the very definition of v = wy , we get that
v ∈ C α (C), and so
u ∈ C α (Ω).
Now, let us consider
C+ = u ∈ C α (Ω) : u(x) ≥ 0 ∀x ∈ Ω ,
whose interior set is
int C+ = u ∈ C α (Ω) : u(x) > 0 ∀x ∈ Ω .
√
Let e1 be the first eigenfunction of −∆ with associated first eigenvalue λ∗
and L2 weight β, i.e. a solution of problem (Pλ∗ ). We first prove that e1 does
exist. In fact, we have
1/2
Proposition 3.4. If H(β) holds, then there exists e1 ∈ H0 (Ω) ∩ int C+ which
is the first eigenfunction with associated eigenvalue λ∗ of problem (Pλ∗ ).
Proof. Let us consider the functional
J(u) =
1
2
Z
|Du|2 dxdy
C
constrained on the set
M=
u∈
Z
1
H0,L
(C)
βu dx = 1 ,
2
Ω
and note that it is sequentially weakly lower semicontinuous and coercive; so by
1
the Weierstrass Theorem we can find e1 ∈ H0,L
(C) such that
n
o
1
J(e1 ) = inf J(u)| u ∈ H0,L
(C) .
R
1
(C),
Thus, there exists λ∗ ∈ R such that J 0 (e1 )v = λ∗ Ω βe1 vdx for every v ∈ H0,L
that is e1 is the first eigenfunction for the problem
(
∆e1 = 0
in C,
∂e1
∗
in Ω.
∂ν = λ βe1
Moreover, it is clear that we can assume e1 ≥ 0 in Ω, and by Proposition 3.3 we
get that e1 ∈ C+ .
Finally, by Remark 3.2, e1 ∈ int C + .
In general no other properties can be deduced for e1 if no other hypotheses on
β are given. However, we can recover the classical features of the first eigenvalue
with additional assumptions. In particular, from Theorem 2.1.4-5 we have the
following regularity result:
Proposition 3.5. If β ∈ C 1,α (Ω), then e1 ∈ C 2,α (Ω).
Logistic square root of the Laplacian with indefinite weight
4
11
The bifurcation theorem
On the absorption term g we assume that
H(g): g : Ω×R −→ R is a Carathéodory function such that g(x, s) = 0 for a.e. x ∈
Ω and for all s ≤ 0 and g(x, s) ≥ 0 for a.e. x ∈ Ω and for all s > 0. Moreover:
1. there exists g0 ∈ L∞ (Ω) such that:
lim inf
s→∞
g(x, s)
≥ g0 (x)
s
uniformly for a.e. x ∈ Ω
(17)
and
ess inf (g0 − β) = µ > 0.
Ω
2.
lim+
s→0
g(x, s)
=0
s
uniformly for a.e. x ∈ Ω
(18)
(19)
3. for every u ∈ L∞ (Ω) there exists ρ = ρ(u) ≥ 0 such that
|g(x, u(x))| ≤ ρ for a.e. x ∈ Ω.
(20)
4.
For a.e. x ∈ Ω the function u 7→
g(x, u)
is strictly increasing in (0, ∞).
u
(21)
Remark 4.1. Hypothesis (17) does not exclude the classical case, that is
lim inf
s→∞
g(x, s)
= ∞,
s
as it happens, for example, if g(x, s) = |s|p−2 s, p > 2.
Remark 4.2. Hypothesis (20) is very general and does exclude any a priori
bound on the growth of the function at infinity, as it is usual in variational
problems of this type.
We shall prove the following result:
Proposition 4.1. If hypotheses H(β), H(g), (17), (18), (19) and (20) hold,
then for all λ > λ∗ there exists a solution u ∈ C α (Ω̄) of problem (Pλβ ) such that
u(x) > 0 for all x ∈ Ω.
Proof. By virtue of hypothesis (17), it follows that for all ∈ (0, µ) there exists
M = M () > 0 such that for all s > M
g(x, s) ≥ (g0 (x) − )s
for a.e.
x ∈ Ω.
(22)
Now we fix ψ > M and we consider the following truncation for the reaction
term:


if s ≤ 0
0
h(x, s) = β(x)s − g(x, s) if 0 ≤ s ≤ ψ
(23)


β(x)ψ − g(x, ψ) if s ≥ ψ.
Logistic square root of the Laplacian with indefinite weight
Of course, h is still a Carathéodory function. Setting H(x, s) =
1
H(g) the functional Iψ : H0,L
(C) −→ R defined by
1
2
Iψ (v) = kDvkL2 (C) − λ
2
12
Rs
0
h(x, s)ds, by
Z
H(x, v(x)) dx
Ω
is of class C 1 and sequentially weakly lower semicontinuous. Moreover, by the
very definition of the truncation h(x, s), we have that Iψ is coercive. Therefore,
1
(C) such that
by the Weierstrass Theorem we can find v ∈ H0,L
1
Iψ (v) = inf Iψ (v) | v ∈ H0,L
(C) = m,
(24)
that is m is a critical value for Iψ .
Now we must check that v 6= 0. By virtue of hypothesis (19), in correspondence of the previous there exists δ = δ() > 0 with δ < min{M, ψ} such
that
g(x, s) ≤ s ∀s ∈ [0, δ] and a.e. x ∈ Ω,
so that
Z
s
g(x, s) ds ≤ G(x, s) =
0
s2
∀s ∈ [0, δ] and a.e. x ∈ Ω.
2
(25)
By Proposition 3.4, we can find t ∈ (0, 1) such that te1 ∈ [0, δ] for all x ∈ Ω.
Then, by (25)
Z
t2
Iψ (te1 ) = ||De1 ||22 − λ
H(x, te1 )dx
2
Ω
Z
Z
t2
t2
2
= kDe1 k2 − λ
βe21 dx + λ
G(x, te1 ) dx
2
2 Ω
Ω
Z
t2
t2
λ
2
2
≤ kDe1 k2 − λ
βe21 dx + t2 ke1 k2
2
2 Ω
2
Z
t2
λ
2
= (λ∗ − λ)
βe21 dx + t2 ke1 k2 .
2
2
Ω
R
Recalling that Ω βe21 dx = 1 and λ∗ < λ, choosing > 0 sufficiently small, then
we have Iψ (te1 ) < 0 and by (24) we find that
m = Iψ (v) < Iψ (0) = 0,
(26)
that is v 6= 0, as claimed.
Since v is a critical point for Iψ then Iψ0 (v̄) = 0, that is
A(v) = λh(x, v),
1
1
where A : H0,L
(C) −→ (H0,L
(C))0 is the linear map defined by
Z
Du · Dv dxdy
(A(u), v) =
C
for all u, v ∈ H01 (C).
(27)
Logistic square root of the Laplacian with indefinite weight
13
1
(C), obtaining
Now, we act on (27) with −(v)− ∈ H0,L
Z
D(v)− 2 dxdy = 0,
(A(v), −(v)− ) =
C
so that v − = 0, that is v ≥ 0.
1
Next we act on (27) with (v̄ − ψ)+ ∈ H0,L
(C) and by definition of h we find
Z
h(x, v̄)(v − ψ)+ dx =
(A(v), (v − ψ)+ ) = λ
Ω
Z
= (βv̄ − g(x, v̄))(v − ψ)+ dx ≤
(28)
Ω
Z
≤ (βψ − g(x, ψ))(v − ψ)+ dx.
Ω
First, let us prove that
Z
(βψ − g(x, ψ))(v − ψ)+ dx ≤ 0.
(29)
Ω
Indeed, recalling that ψ > M and < µ, using (22), by (18) we have
βψ − g(x, ψ) ≤ βψ − g0 (x)ψ + ψ ≤ ( − µ)ψ < 0.
(30)
Then, by (28),
Z
|Dv̄|2 dxdy ≤ 0,
{x∈C : v̄>ψ}
so that, from (28), (29) and (30), we find that
(βψ − g(x, ψ))(v − ψ)+ = 0 in Ω.
Using again (30), this implies that (v − ψ)+ = 0 in Ω, that is v̄ ≤ ψ in Ω.
Therefore, v is not only a solution of the truncation problem, but also of problem
(Qλ ); then ū = trΩ v̄ is a solution of problem (P ). Finally, (20) and elliptic
regularity for Neumann problems in Lipschitz domains (for instance, see [18])
¯ and so u ∈ C α (Ω̄).
imply that v ∈ C α (C),
Proposition 4.2. If hypotheses H(β) and (21) hold, then the nontrivial nonnegative solution of problem (Pλβ ) is unique.
Proof. We consider two nontrivial nonnegative solutions u, v ∈ C α (Ω) of problem
2
2
−v 2
−u2
1
Pλβ and we test with uu+
and v v+
, which clearly belong to H0,L
(C). Then
u2 − v 2
dxdy = λ
u+
Z
v 2 − u2
Dv · D
dxdy = λ
v+
C
Z
Z
Du · D
C
Z
Now
D
u2 − v 2
dx − λ
u+
Z
v 2 − u2
βv
dx − λ
v+
Ω
Z
βu
Ω
u2 − v 2
dx
u+
(31)
v 2 − u2
dx.
v+
(32)
g(x, u)
Ω
g(x, v)
Ω
u2 − v 2
u2 + 2u
2vDv
v2
=
Du
−
+
Du,
u+
(u + )2
u+
(u + )2
(33)
Logistic square root of the Laplacian with indefinite weight
2
14
2
−u
and an analogous form for D v u+
.
Adding (31) and (32), we find
Z
Z
u2 − v 2
v 2 − u2
Du · D
dxdy + Dv · D
dxdy
u+
u+
C
C
3
Z
vu2
u3
v2 u
v
−
+
−
dx
=λ
β
v+ v+ u+ u+
Ω
Z
Z
u2 − v 2
v 2 − u2
−λ
dx − λ
dx.
g(x, u)
g(x, v)
u+
v+
Ω
Ω
By (33) this reads as
2
Z
2vDv
u + 2u
v2
Du
−
Du
dxdy
0=
Du ·
+
(u + )2
u+
(u + )2
C
2
Z
2uDu
u2
v + 2v
+ Dv
Dv
−
+
Dv
dxdy
(v + )2
v+
(v + )2
C
Z
Z
u2 − v 2
v 2 − u2
+λ
g(x, u)
dx + λ
g(x, v)
dx
u+
v+
Ω
Ω
3
Z
v
vu2
u3
v2 u
−λ
β
−
+
−
dx
v+ v+ u+ u+
Ω
Z
=
|D log(u + )|2 + |D log(v + )|2 − 2 hD log(u + ), D log(v + )i (u2 + v 2 )dxdy
C
Z
Z
u2 − v 2
v 2 − u2
+λ
g(x, u)
dx + λ
g(x, v)
dx
u+
v+
Ω
Ω
3
Z
v
vu2
u3
v2 u
−λ
β
−
+
−
dx
v+ v+ u+ u+
ZΩ h
i
+ 2
|D log(u + )|2 u + |D log(v + )|2 v − (u + v)hD log(v + ), D log(u + )i dxdy,
C
i.e.
Z
|D log(u + ) − D log(v + )|2 (u2 + v 2 )dxdy
C
Z g(x, u) g(x, v)
+λ
−
(u2 − v 2 )dx
u
+
v
+
ZΩ h
i
+ 2
|D log(u + )|2 u + |D log(v + )|2 v − (u + v)hD log(v + ), D log(u + )i dxdy
C
3
Z
v
vu2
u3
v2 u
−
+
−
dx = 0.
−λ
β
v+ v+ u+ u+
Ω
(34)
Now, by (20) we have
g(x, u) g(x, v)
ρ(kuk∞ ) ρ(kvk∞ )
2
2 −
(u
−
v
)
+
|u2 − v 2 | ∈ L1 (Ω),
≤
u+
v+
kuk∞
kvk∞
and by the Lebesgue Theorem we immediately find that
Z Z g(x, u) g(x, v)
g(x, u) g(x, v)
2
2
−
(u − v )dx −→
−
(u2 − v 2 )dx
u+
v+
u
v
Ω
Ω
Logistic square root of the Laplacian with indefinite weight
15
as → 0.
Second,
Z h
i
|D log(u + )|2 u + |D log(v + )|2 v − (u + v)hD log(v + ), D log(u + )i dxdy 2 C
|Du|2
|Dv|2
|Du||Dv|
≤ 2
+
+
(u + )2
(v + )2
(u + )(v + )
2
2
|Du|
|Dv|
≤
+
+ |Du||Dv| ∈ L1 (C),
2
2
and again the Lebesgue Theorem implies that the integral under consideration
tends to 0 as → 0.
Moreover, it is easy to see that, similarly,
3
Z
vu2
u3
v2 u
v
−
+
−
dx → 0
β
v+ v+ u+ u+
Ω
as → 0.
In conclusion, passing to the lim inf in (34), Fatou’s Lemma implies that
Z
Z g(x, u) g(x, v)
2 2
2
0≥
|D log(u)−D log(v)| (u +v )dxdy +λ
−
(u2 −v 2 )dx.
u
v
C
Ω
Since both integrals are nonnegative by (21), we immediately get that u = v, as
desired.
Next we prove that:
Proposition 4.3. If hypotheses H(β) and H(g) hold, problem (Pλβ ) has no positive solutions when λ ∈ (0, λ∗ ] .
Proof. Suppose that there exists a positive solution v0 =
6 0 for (Qβλ ), that is
Z
Z
1
Dv0 · Dv dxdy = λ [βv0 − g(x, v0 )]v dx ∀ v ∈ H0,L
(C).
C
Ω
In particular
Z
Z
2
[βv02 − g(x, v0 )v0 ]dx < λ
|Dv0 | dxdy = λ
C
Ω
Z
βv02 dx
Ω
Comparing with the definition of λ∗ , we have
Z
Z
Z
2
∗
2
λ
βv0 ≤
|Dv0 | dxdy < λ
βv02 dx,
Ω
C
Ω
∗
and we reach a contradiction, since λ ≤ λ .
In conclusion, we can state the following theorem:
Theorem 4.1. If hypotheses H(β), H(g), (17), (18), (19), (20), (21) hold, then
there exists λ∗ > 0 such that
Logistic square root of the Laplacian with indefinite weight
16
1. for all λ > λ∗ problem (Pλβ ) has a unique positive solution u ∈ C α (Ω).
2. for all λ ≤ λ∗ problem (Pλβ ) has no positive solutions.
Finally, we concentrate on the behaviour of solutions near λ∗ , i.e. we concentrate on
lim∗+ u(λ).
λ→λ
For this purpose, we suppose that there exist p ∈ (2, 2] ) and C0 > 0 such that
g(x, s) ≥ C0 |s|p−1 for a.e x ∈ Ω and for all s ∈ R.
(35)
Then we have
p
Proposition 4.4. Assume that H(g) holds and that β + ∈ L p−2 (Ω) with β + 6= 0.
If hypothesis (35) holds, then every solution uλ of problem (Pλβ ) converges to 0
1
in H 2 (Ω) when λ ↓ λ∗ .
Proof. Let (λn )n be a sequence strictly decreasing to λ∗ , and let vn be a corresponding solution of problem (Qβλn ) (notice that under these assumptions no
uniqueness is granted).
Then
Z
Z
2
|Dvn | dxdy = λn [βvn − g(x, vn )]vn dx ∀n ∈ N.
(36)
C
Ω
By Lemma 2.2, the Hölder inequality and (35), we have that
Z
Z
Z
C||vn ||2p ≤
|Dvn |2 dxdy =λn
β|vn |2 dx − λn
g(x, vn )vn dx ≤
C
Ω
Ω
+
≤λn (||β ||
0
(p
2)
||vn ||2p
− C0 ||vn ||pp ).
Being p > 2, the sequence (vn )n is bounded in Lp (Ω), and therefore in L2 (Ω).
1
(C).
As a consequence, by (36), (vn )n is bounded in H0,L
Thus, up to a subsequence, we have that
1
vn * v ∗ in H0,L
(C) and vn → v ∗ in Lp (Ω) as n → ∞.
1
Using the definition of solution for vn with vn − v ∗ ∈ H0,L
(C) as test function,
we get
Z
Z
Z
Dvn ·D(vn −v ∗ )dxdy = λn
βvn (vn −v ∗ )dx−λn
g(x, vn )vn (vn −v ∗ )dx → 0
C
Ω
Ω
as n → ∞, so that
Z
lim
n→∞
C
|Dvn |2 dxdy =
Z
|Dv ∗ |2 dxdy,
C
that is
1
vn → v ∗ in H0,L
(C).
Logistic square root of the Laplacian with indefinite weight
17
Setting un = T rΩ vn and u∗ = T rΩ v ∗ , exploiting the continuous injection of
1
1
T rΩ (H0,L
(C)) ⊂ H 2 (Ω), we obtain that
1
un → u∗ in H 2 (Ω).
Now, observe that
p 0
p
> N,
=
p−2
2
since 2 < p < 2] . Then, by Proposition (4.3), we find u∗ = 0.
Remark 4.3. Let us remark that in the previous case, in agreement with H(β),
p
> N , since p < 2] .
we have q = p−2
In the light of Proposition 4.4, we emphasize the fact that if g and β satisfy all
the assumptions above, we obtain a complete bifurcation result for our problem
that we summarize in the following final
p
Theorem 4.2. Assume that β ∈ L p−2 (Ω) with β + 6= 0. Moreover, assume H(g)
and (17)–(21) and (35). Then the conclusions of Theorem 4.1 and Proposition
4.4 hold true.
Acknowledgements
The authors are member of the Gruppo Nazionale per l’Analisi Matematica, la
Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta
Matematica (INdAM), and are supported by the GNAMPA Project Systems with
irregular operators.
Finally, the authors sincerely thank the anonymous referees for their careful
checking of the paper and for their useful remarks, which improved the presentation of the paper itself.
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