Uniqueness and continuum of foliated
~. solutions for a quasilinear elliptic equation
with a non lipschitz nonlinearity
G.Barles
Dpt. Mathematiques et Informatique Scientifique.
Faculte des Sciences et Techniques. Universite de Tours.
Parc de Grandmont. 37200- Tours. France.
G.Diaz * and
J.I.Diaz *
Dpt. Matematica Aplicada. Facultad de Matematicas.
Universidad Complutense de Madrid. 28040-Madrid. Spain.
where fl is a smooth bounded domain in RN, oX ;::: 0 and H is a given
continuous function.
It is well-known that (1) satisfies the classical Maximum Principle in
C2(fl) n C(fi), provided that oX > O. This result remains true when oX = 0 if
H(x,p) is assumed to be locally Lipschitz continuous in p, by using Hopf's
Maximum Principle (c.f. [4J or [5]). The aim of this paper is to investigate
the
case
when A
= 0 and
when we drop the locally Lipschitz assumption on
we will consider only the case when
p. To simplify the presentation,
where 0 < m < 1 and f E co·<>(n).
We will be interested first in existence results. With A = 0 and H being
given by (2), the equation (1) becomes
provided t.p E C(80).
Concerning the existence of a solution we have
Theorem
1 Assume that 0 is a C2,/3 domain for some 0 < f3 ~ I, then the
problem (3) and (4) has a minimal u and a maximal solution U E C2,6(n) n
C(O) for some S depending on a, f3 and m.
Of course, u and U can be different in general, as we will show it by several
examples. But uniqueness still holds under some additional assumptions. So,
a first result is the following
Theorem
2 Assume that f(x) =f:. 0, \Ix E n. Then, the Maximum Principle
holds for (3) in C2(n) nC(n). In particular, the solution of (3)-(4) is unique.
Then, the natural question is to know whether we can relax or not the
assumption on f in Theorem 2. It is a curious fact that the answer will
depend on the sign of f. Indeed, we will give an example of function f, f > 0
in n\{xo} and f(xo)
0 for some Xo E n for which (3)-(4) exhibits three
solutions. By the contrary, we shall show that uniqueness still remains true
if f ~ 0 and the set where f vanishes is small enough. As a matter of fact,
our answer for this last case only concerns with radially symmetric solutions
of (3)-(4) on a ball. Finally, we shall prove that there exists a continuum of
solutions for the Dirichlet problem, whenever uniqueness fails. In fact, this
set generates, near a point where f vanishes, a foliation of dimension N and
codimension 1.
The paper is organized as follows: Section 2 is devoted to the proof of
Theorem 1 and also contains a first example of non "iqueness for the radial
case. Theorem 2 will be proved in Section 3. Finally, in Section 4 we study
the case when f may vanish and we give uniqueness and non uniqueness
results according the sign of f. We end the paper by studying the structure
of the set of solutions in absence of uniqueness.
=
We first give the proof of Theorem 1. We start by considering the case when
rp E 2.-r(on) for some I> O. We approximate (3) and (4) by the problems
c
(~):
-~u.
+ (t2+
=f
2
I VUe 1
)T
= rp
u(
-~U(
+ (t + I VUe
2
/2)T -
E.2
=f
U. = rp
inn,
on on,
in n,
on
on.
By classical results (c,J. [4]), (E.) and {P.) have unique solutions since, now,
the nonlinearities are locally Lipschitz continuous. Then, due to 0 < m < 1,
using Schauder estimates, one proves easily that
U(
-t
in C2(!l),
U
and u, U are both C2.S(!l) solutions of (3)-(4).
Moreover, if we point out the dependence of u and U in the boundary
datun rp by denoting them respectively u(r.p), U(rp), one has
=
for v = u, U and rp, t/J E C2"·Y(on). Indeed, (5) is true for v
u. (or v = U.)
and we pass to the limit in the inequality. In order to solve (3)-(4) for
rp E C(on), we introduce two sequences {!e.}, {<p.} such that:
1.
!e. $. rp $. <Pc
2.
!e.,<P.
3. !e.,lp(
E c2·'Y(on)
-t
rp
for some I E]O, 1[,
in C(on)
when
Using (5), it is clear enough that
in C(!l) and therefore
u(!e.)
U(<p.)
t
-t
O.
{U(lp.)} and {u(!e.)} are Cauchy sequences
-t
u
in C(TI)
-t
U
in C(!l).
Moreover, u(!e.) and U(<p.) satisfy uniform interior estimates in C2.S(!l) (for
some 6 depending on a,f3 and m) and therefore u and U solve (3).
Finally, let w be a solution of (3)-(4) wE C2(n) n C(fi). Then using the
above notations and the Maximum Principle for (£.) and {Pt), one has
u:S;w:S; U
on
fi.
Therefore, u is the minimal solution of (3)-(4) and U is the maximal solution
of (3)-(4) and the proof is complete.#
Now, we turn to a first example of non-uniqueness in the radial case.
Example 1.
We consider the boundary problem
-D.u+
I
Vu
1m
u
=0
in B(O, R),
on aB(O,R)
=c
where B(O, R) is the open ball in RN of radius R centered at the origin and
c E R. Of course, u == c is a solution of (6). Easy computations show that
the function
2-m
k=--
I-m
Function w given by (7), or a modification of it, is very useful in order to
show the existence and localization of a free boundary for the problem
-D.u+
I
Vu
1m
+,\u
u
=f
= cp
in n,
on an.
Clearly, the first order nonlinearity becomes, near the set {x : I Vu 1= O},
the dominant term of the PDE. In fact, this set is not empty provided suitable conditions on cp, f and n. Indeed, assume, for instance f E CO,"'(n), cp E
C(an) and ,\ > O. Standard arguments lead to the existence of a unique solution u E C2(n) n C(fi) of (8).
Theorem 3 Let us suppose that there exists c E R such that
that
f ;::c
in nand
c
(Il>
A
T-
-
and Let u be the soLution of (8). Then
c
u(xo)
= 'A
dist (xo, supp (<p -
(I»
for any Xo E Nc(f)
such that
R
= (A II u 11£00(0)
ACm
~ R
U f)Nc(f»
-c] t
for k and Cm as in Example 1.
Proof.
The Maximum Principle and assumptions imply
u( x) ~
Let Xo E Nc(f)
c
'A'
xE
-
n.
satisfying (9). Then the function
c
uc(x) = ~ + Cm I x - Xo
is a supersolution
fi c Nc(f)).
Since
of the PDE on
k
I
n = B(xo, R) n n
(notice the inclusion
The boundary of the set {x: u = f} is a free boundary whose study can
be carryed out following the techniques of [3]. The above result shows that
the behaviour of the solutions of (8) is very close to the one of solutions of
elliptic equations degenerating on the set {'Vu
o} such as, for instance,
=
-
div (I 'Vu IP-2 'Vu)
+ Au = f,
p> 2
(10)
(see [3, Theorem 1.14)).
Finally, we pointed out that when c f:. 0 all those free boundaries disappear if A -+ O. Moreover, if A = 0, f::; 0 the Strong Maximum Principle for
nonnegative subharmonic functions shows that any solution u of (3) verifies
u < 0 in n, unless u == O.
3
Uniqueness for non vanishing right side
terms.
We give in this part the proof of Theorem 2. Since f is continuous in n it
is enough to prove the uniqueness for f < 0 or f > O. We first consider the
case when f < 0 in n. The idea is to make some suitable change of variable
u ~(v), where ~ is a strictly monotone smooth function. After this change,
(3) becomes
=
_ L).v - cI>"(V) I 'Vv
~'(v)
12
+(cI>'(v»m-l
I
'VV 1m -(~'(v»-lf
=0
in n. (11)
Since we deal with smooth solutions, the only property to be checked is that
the nonlinearity is strictly increasing with respect to v, provided we make a
suitable choice of ~. We choose
and the desired property is obviously satisfied since 1 - m > 0 and f < O.
Now, we turn to the case when f > 0 in n. We are going to give two proofs
in this case. The first one follows exactly the idea explained just above:
unfortunately, the choice of cI>is not obvious anymore. So, we turn to the
computations made in [2) which say that v has to be defined through the
relation
loo
ds
U(.,)
M
- exp
{•-as }
= v(x),
x E
-
n,
where M, A are large constants chosen later. Constant M has, in particular,
to be chosen large enough in order to ensure that the above integral makes
sense. This means that the function ~ satisfies the ODE
H(x,t,p)
=-
::gj
I
p
12
+(cI>'(t»m-l
I
p 1m -(cI>'(t»-l f(x),
aH(x,t,p)
at
=
_(iI>//(t»)'
iI>'(t)
I
(:':i~])'=
p
12
+(m -1)iI>//(t)(iI>'(t»m-21
p
1m
+ iI>//(t)f(x).
iI>'(t)
-A2exp{-AiI>(v)}iI>'(v)
= -AiI>//(v).
Since iI>//> 0, we are just interested in the sign of the bracket. Using Young
inequality, we have
(I - m)(iI>'(v»m-2 I p 1m
m
2- m
5 "2A I p 12 +-2-A~(1-
-2....
-m
m)2-m(iI>'(v»-
2
and the bracket is estimed by
m
A(1-"2)lpI2+(iI>'(v»
2 [
2- m
.=m.
f(x)--2-A2-m(l-m)2-m.
-2....]
To conclude, it is suffices to choose A large in order to have the last term
strictly positive and then M is chosen verifying
and the proof is completed.
We give now another proof of the above result. We consider a subsolution
wand a supersolution W of (3); we assume w, WE C2(n)nC(f!) and w 5 W
on an. Our aim ·is to show that w 5 W on f!. To do so, we argue by
contradiction assuming that max (w - W) > O. This implies, in particular,
that no maximum point of w - W lies on an. Moreover, if Xo E n is a
maximum point of w - W, then
Indeed, if Vw(xo)
=/: 0, the
contradiction
follows from the fact that the Hopf
Principle applies, since in a neighborhood of Xo, functions wand Ware
solutions of a quasilinear elliptic equation with a Lipschitz nonlinearity.
that, we consider the function
So
This function has at least a maximum point xI' and (taking if necessary a
subsequence) we may assume that
where
yield
Xo
is a maximum point of
W -
W. Properties of the interior maxima
J.l.6.W(XIJ) ::; 6.W(xlJ),
J.I.'Vw(xlJ)
But wand
= 'VW(xlJ)'
Ware respectively sub and supersolutions of (3); hence
-Aw(xlJ)+
'Vw(XIJ)
I
m
1 ::;
f(xlJ),
I 'VW(XIJ) Im~ f(xl')'
-AW(xlJ)+
Then, we multiply the first inequality by Il, we subtract the second one and
using the properties above we are led to
which is the desired contradiction
since 'VW(xo)
= 0 and
f(xo)
> 0.#
Remark l.
A slight modification of the above arguments gives also another proof for
the" f < 0" case.#
4
The case of
f vanishing at one point.
It is natural to wonder whether Theorem 2 is sharp or if one can relax the
assumptions by allowing, for example, f to vanish at a point or (Ill some
"small" subset of
We first build an example showing that the" f > 0"
result is sharp.
n.
Example 2.
For f(x) = v
I X Il.:'m, v>
-Au+
I
0, c
'Vu
E R,
1m
u
we consider the boundary problem
=f
=c
in B(O, R),
on 8B(0,R)
[ 2-m
g(C) = (1- m)2
Easy computations
+N
-1
]C + (2-m)m
1- m
IC
m
1 ,
C E R.
show that u solves (13) if and only if and only if
2-m
k=-1-m
But if II is small enough, this algebraic equation has two negative roots and
one positive which lead to three different radially symmetric solutions for
(13). We emphasize that f > 0 in B(O, R)\ {O} and 1(0) = O. We also
remark that, in fact, two of those solutions are negative solutions although
f;::: 0.#
Our next result shows that the situation changes when f is nonpositive
and the set where f vanishes is small enough. We only consider the case of
radially symmetric solutions
n
Then, for
= B(O, R) the problem (3)-(4) has at most one radially symmetric solution.
Proof.
Let v(x)
v(1 x I) be any radially symmetric
n B(O, R) and define
=
=
w(r)
- w'(r)
= dv
dr (r)rN-1,
= rN-1f(r)
- r(N-l)(l-m)
0
I
solution of (3)-(4) on
< r < R.
w(r)
1m,
0
< r < R.
(15)
So, w'(r) > 0 a.e. r E]O,R[. Moreover, as w(O) = 0, we have that w(r) > 0
for any r E]O,R[ (notice that w ECl([O, R[». Then, we can define the inverse
function r = a(t) (i.e. w(a(t» = t). From (15) we deduce that
1
bet) = -1
(We note that q + 1
q = (N - 1)(1
- m).
a(t)q+l,
q+
> 0 and w(O) = 0 implies b(O) = 0). Then, we have
Notice that P(TJ) < 0 a.e. TJ > O.
At this point, assume that (3)-(4) has two different radially symmetric
solutions UI and U2' They generate two solutions bt(t) and b2(t) of (14)
with bI =I- b2• Since bt (0) = b2(0), one has
bt(t)
= b (t),
2
if 0:5 t :5 t
II bI
-
b2
and
IILOO(t,T)=
(bI
-
b2)(s)
Integrating by parts in (16) in It, sl, properties .r(0)
lead to
> O.
= 0 and
which contradicts that .r is strictly decreasing. Then, bI
W2 in ]O,R[. In particular,
u~(r) = u~(r) in 10,R[ and as
ui(r)
= cp
-l
R
uHs)ds,
if t. <t :5 T
bt(t) =I- b2(t),
0
<r <R
(i
(bI
== b2
= 1,2)
-
b2)(t)
and so
=0
WI
==
Remark 2.
We do not know, update, if the above result can be extended
general (non radially symmetric) case for functions f satisfying f
n\{xo}, f(xo)
0, for some Xo E n.#
=
to the
< 0 in
We conclude this paper by studying the structure of the set of solutions
provided that the following condition holds:
there exists Xo E n such that f(xo)
= 0 and f(x) =/: 0 Vx
E n\{xo}.
(17)
Theorem 5 Assume (17). Then, for any YEn and any t E [u(y), U(y)]
there exists a solution U of(3)-(4) in C2(n)nC(n) satisfying u(y) = t, where
U and U denote respectively the minimal and the maximal solution of(3)-(4).
Proof.
First step.
We start by proving the result for the special case y
be defined by
t = u(xo) + tt = U(xo) - t2.
= xo.
Let tt, t2
>0
We denote by Ut(.) the function u(.)+tt and by U2(') the function U(.) -t2.
We will proceed in the following way: we will first prove that there exists a
unique solution Ut E 2'l'(n\{xo}) n C(TI) of
c
-~u+
I
Vu 1m
u(xo)
= t,
=f
in n\{xo},
U
= <p
on
an,
and then we will show that Ut E c2,'l'(n) and that the equation holds at xo.
To do so, we introduce the approximated problem
-~U.+
I
Vu.
1m
=f
u.
= U2
u. = <p
in n\B(xo, e),
on aB(xo, e),
on
an.
where e is small enough. By Theorem 1 and 2, there exists a unique solution
u. of (19) in 2''l'(n\B(xo, e)) n C(n\B(xo, e)). The aim is to pass to the
limit in (19). The first ingredient is, of course, the uniform interior estimates
for u. in 2,y(n\{xo}) for some 0 < 'Y < 1. Then we have to work in a
neighborhood of the boundaries. We first claim that
c
c
indeed, Ut and U2 are solutions of (3), Ut(xo) = U2(XO) and Ut ;:::U2 on
(20) is a consequence of Theorem 2 since f does not vanishes in
n\{xo}. By the same argument, one has also
an; hence
By the uniform interior estimates, we can consider a sequence {U.,}., converging in C2(n\{xo}) to some function Ut satisfying
-~Ut+
I V7Ut Im= f
in n\{xo},
in n\{xo},
U2 ~ Ut ~ U1
U ~Ut
~
U
on
fl.
Therefore Ut E C(11), Ut(xo) = t, Ut = <pon on. We still have to obtain
the equation at Xo. Recalling the arguments of the proof of Theorem 2, we
know that if u ::f: U, Xo is the only maximum point of U - u on fl, moreover
Vu(xo) = V7U(xo) = O. Since
D2U(xo) ~ D2u(xo)
(indeed a nonpositive symmetric matrix with a zero trace is clearly the zero
matrix). Therefore, (23) implies that Ut is twice differentiable at Xo and
And finally this implies that the equation holds at xo, that Ut E C1(n) and
classical regularity results show that Ut E c2'Y(n). Uniqueness of Ut follows
from Theorem 2.
Second step.
We still denote by Ut the solution of (3)-(4) built in the above step such
that Ut(xo) = t, for t E [u(xo), U(xo)). We claim that for any yEn
the
function
t
H
Ut(Y)
is continuous; consequently, the set {Ut(Y) : t E [u(xo), U(xo)]} is a nonempty
subinterval which is necessarily equals to [u(y), U(y)), whence the result follows. Indeed, {Ut} is relatively compact in C2,"I(n) and when t -t T, Ut
converges necessarily to UTI because if the subsequence Uti -t V in C2(n), v
is a solution of the equation in n\ {xo} and v(xo) = Tj therefore v = UT by
uniqueness in n\{xo}. In particular,
and the claim is proved.#
Remark 3.
From the above proof it is easy to see that for t,s E [u(xo), U(xo)] with
t < s implies
Ut(X) < u$(x),
provided
I VUt(x) I + I VU$(x) I> 0
or x close enough xo,#
Theorem 5 shows that the set of solutions generates a compact N
dimensional manifold
Theorem 6 There exists a positive constant 0' such that Nn (B(xo,
is a foliation of dimension Nand codimension 1.
0')
+1
x R)
Proof.
Theorem 5 implies the relation
Moreover, from the compactness of the interval Iu(xo), U(xo)] there exists
> 0 such that the sets Nt are connected and disjoints when they intersect
the cylinder B(xo, 0') x R. As Nt are N-dimensional manifolds, it is easy to
check that for any (Yo,to) EN n (B(xo, 0') x R) there exist two "small" balls
B(yo,o) C B(xo,O') and B(zo,,), a positive constant T and a map
0'
{)t:
B(yo, 0) x ]to - T, to + T[.•..• RN+1
So, according of [1, Definition 4.4.4] the set
of dimension Nand codimension 1.#
N n (B(xo, 0') x R) is a foliation
[11 R.Abraham, J.E.Marsden and T.Ratiu: Manifolds, Tensor Analysis, and
Applications. (2nd ed.) Springer-Verlag. 1988.
[2] G.Barles and F.Murat: In preparation.
[3] J.I.Diaz: Nonlinear Partial Differential Equations and Free Boundaries.
Vol.!. Elliptic Equations. Research Notes in Mathematics nOl06. Pitman. 1985.
[4] D.Gilbarg and N.S.Trudinger: Elliptic Partial Differential Equations of
Second Order. (2nd ed.) Springer-Verlag. 1983.
[5] P.L.Lions:
Quelques remarques
sur les problemes elliptiques
quasilimlaires du second ordre. J. d'Analyse Mathematique.
45
(1985),234-254.