On a nonlocal multivalued problem in an Orlicz-Sobolev space via Krasnoselskii's genus

0.21 MB·English

by  Giovany M. Figueiredo

On a nonlocal multivalued problem in an
5
Orlicz-Sobolev space via Krasnoselskii’s genus
1
0
2
n Giovany M. Figueiredo∗
a Universidade Federal do Para´, Faculdade de Matem´atica,
J
CEP: 66075-110, Bel´em - Pa, Brazil
2
2 e-mail: [email protected]

]
P Jefferson A. Santos †
A Universidade Federal de Campina Grande,
. Unidade Acadˆemica de Matem´atica e Estat´ıstica,
h
t CEP:58109-970, Campina Grande - PB, Brazil
a
m e-mail: jeff[email protected]
[

1
v
6
9
5
Abstract
5
0
.
1 This paper is concerned with the multiplicity of nontrivial solutions
0 in an Orlicz-Sobolev space for a nonlocal problem involving N-functions
5 and theory of locally Lispchitz continuousfunctionals. More precisely, in
1
this paper, we study a result of multiplicity to the following multivalued
:
v elliptic problem:
i
X
r
−M Φ(|∇u|)dx div φ(|∇u|)∇u −φ(|u|)u∈∂F(u) in Ω,
a  u∈W(cid:18)Z01ΩLΦ(Ω), (cid:19) (cid:0) (cid:1)

where Ω ⊂ RN is a bounded smooth domain, N ≥ 2, M is continuous
|t|
function, Φ is an N-function with Φ(t) = φ(s)s ds and ∂F(t) is a
Z0
generalized gradient of F(t). We use genus theory to obtain the main
result.
∗PartiallysupportedbyCNPq/PQ301242/2011-9 and200237/2012-8
†PartiallysupportedbyCNPq-BrazilgrantCasadinho/Procad552.464/2011-2
1
2
1 Introduction
The purpose of this article is investigate the multiplicity of nontrivial solutions
to the multivalued elliptic problem
−M Φ(|∇u|)dx div φ(|∇u|)∇u −φ(|u|)u∈∂F(u) in Ω,
(P)
 u∈W(cid:18)0Z1ΩLΦ(Ω), (cid:19) (cid:0) (cid:1)
t
whereΩ⊂RN isaboundedsmoothdomainwithN ≥2,F(t)= f(s)dsand
Z0
∂F(t)= s∈R;F0(t;r)≥sr, r ∈R .
Here F0(t;r) denotes the g(cid:8)eneralized directional deriva(cid:9)tive of t 7→ F(t) in
direction of r, that is,
F(h+sr)−F(h)
F0(t;r)=limsup .
s
h t,s 0
→ ↓
We shall assume in this work that f(t) is locally bounded in R and
f(t)=limess inf{f(s);|s−t|<ǫ} and f(t)=limess sup{f(s);|s−t|<ǫ}.
ǫ 0 ǫ 0
↓ ↓
It is well known that
∂F(t)=[f(t),f(t)], (see [13]),
and that, if f(t) is continuous then ∂F(t)={f(t)}.
Problem (P) with φ(t)=2, that is,
−M |∇u|2 dx ∆u−u∈∂F(u) in Ω,
(∗)
 u∈H(cid:18)1Z(ΩΩ) (cid:19)
 0
is called nonlocal because of the presence of the term M |∇u|2dx which
(cid:18)ZΩ (cid:19)
implies that the equation (∗) is no longer a pointwise identity.
The reader may consult [3], [2], [23] and the references therein, for more
information on nonlocal problems.
On the other hand, in this study, the nonlinearity f can be discontinuous.
Thereisbynowanextensiveliteratureonmultivaluedequationsandwereferthe
readerto [4],[20],[6],[5], [11],andreferencestherein. The interestinthe study
of nonlinear partial differential equations with discontinuous nonlinearities has
increasedbecausemanyfreeboundaryproblemsarisinginmathematicalphysics
may be stated in this form.
Among these problems, we have the obstacle problem, the seepage surface
problem, and the Elenbaas equation, see for example [13], [14] and [15].
For enunciate the main result, we need to give some hypotheses on the
functions M,φ and f.
The hypotheses on the function φ:R+ →R+ of C1 class are the following:
3
(φ ) For all t>0,
1
φ(t)>0 and (φ(t)t) >0.
′
lN
(φ ) There exist l,m∈(1,N), l≤m<l = such that
2 ∗
N −l
φ(t)t2
l≤ ≤m,
Φ(t)
t
| |
for t>0, where Φ(t)= φ(s)sds.
Z0
The hypothesis on the continuous function M :R+ →R+ is the following:
(M ) There exist k ,k ,α,q ,q >0 and b:R→R of C1 class such that
1 0 1 0 1
k tα ≤M(t)≤k tα,
0 1
α> q1, where
l
b(t)t2
m<q ≤ ≤q <l ,
0 1 ∗
B(t)
for all t>0 with
(b(t)t) >0, t>0
′
and
t
B(t)= b(s)sds.
Z0
The hypotheses on the function f :R→R are the following:
(f ) For all t∈R,
1
f(t)=−f(−t).
(f ) There exist b ,b >0 and a ≥0 such that
2 0 1 0
b b(t)t≤f(t)≤b b(t)t, |t|≥a .
0 1 0
(f ) There exists a ≥0 such that
3 0
f(t)=0, |t|≤a .
0
The main result of this paper is:
Theorem 1.1 Assumethat conditions (φ ), (φ ), (M ), (f )−(f )hold. Then
1 2 1 1 3
for a > 0 sufficiently small (or a = 0), the problem (P) has infinitely many
0 0
solutions.
4
Belowweshowtwographsoffunctionsthatsatisfythehypotheses(f )−(f ).
1 3
Note that the second graph corresponds to a function that has an enumerable
number of points of discontinuity.
b1 b(t)t b0 b(t)t b1 b(t)t b0 b(t)t
f(t) f(t)
a a
0 0
In the last twenty years the study on nonlocal problems of the type
−M |∇u|2 dx ∆u=f(x,u) in Ω,
(K)
 u∈H(cid:18)1Z(ΩΩ) (cid:19)
 0
grew exponentially. That was,probably, by the difficulties existing in this class
of problems and that do not appear in the study of local problems, as well as
due to their significance in applications. Without hope of being thorough, we
mention some articles with multiplicity results and that are related with our
main result. We will restrict our comments to the works that have emerged in
the last four years
Theproblem(K)wasstudiedin[23]. Theversionwithp-Laplacianoperator
wasstudiedin[19]. Inbothcases,theauthorsshowedamultiplicityresultusing
genus theory. In [27] the authors showed a multiplicity result for the problem
(K) using the Fountain theorem and the Symmetric Mountain Pass theorem.
In all these articles the nonlinearity is continuous. The case discontinuous was
studied in [20]. With a nonlinearityof the Heaviside type the authorsshoweda
existenceoftwosolutionsviaMountainPassTheoremandEkeland’sVariational
Principle.
Inthisworkweextendthestudiesfoundinthepapersaboveinthefollowing
sense:
a) We cannot use the classical Clark’s Theorem for C1 functional (see [21,
Theorem3.6]),becauseinourcase,theenergyfunctionalisonlylocallyLipschitz
continuous. Thus, in all section 5 we adapt for nondifferentiable functionals an
argument found in [8].
b) Unlike [20], we show a result of multiplicity using genus theory considering
a nonlinearity that can have a number enumerable of discontinuities.
c) Problem (P) possesses more complicated nonlinearities, for example:
5
(i) Φ(t)=tp0 +tp1, 1<p0 <p1 <N and p1 ∈(p0,p∗0).
(ii) Φ(t)=(1+t2)γ −1, γ ∈(1, N ).
N 2
−
(iii) Φ(t)=tplog(1+t) with 1<p0 <p<N −1, where p0 = −1+√21+4N.
(iv) Φ(t)= 0ts1−α sinh−1s βds, 0≤α≤1, β >0.
R (cid:0) (cid:1)
d)We workwithOrlicz-Sobolevspacesandsomedifferentestimatesfromthose
found in the papers above are necessary. For example, the Lemma 3.1 is a
versionfor Orlicz-Sobolevspaces of a well-known result of Chang (see [13], [17]
and [20, Lemma 3.3]). In the Lemma 5.2 one different estimate was necessary
because of the presence of the nonlocal term.
The paper is organized as follows. In the next section we present a brief
reviewonOrlicz-Sobolevspaces. Insection3werecallsomedefinitionsandbasic
results on the critical point theory of locally Lipschitz continuous functionals.
We also present variational tools which we will prove the main result of this
paper. Furthermore,inthischapter,weprovetheLemma3.1,whichisaversion
forOrlicz-Sobolevspacesofawell-knownresultofChang(see[13], [17]and[20,
Lemma 3.3]). In Section 4 we present just some preliminary results involving
genustheorythat willbe usedinthis work. Inthe Section 5we proveTheorem
1.1.
2 A brief review on Orlicz-Sobolev spaces
Let φ be a real-valued function defined [0,∞) and having the following
properties:
a) φ(0)=0, φ(t)>0 if t>0 and lim φ(t)=∞.
t
b) φ is nondecreasing, that is, s>t→i∞mplies φ(s)≥φ(t).
c) φ is right continuous, that is, lim φ(s)=φ(t).
s t+
Then, the real-valued function Φ→defined on R by
t
| |
Φ(t)= φ(s) ds
Z0
is called an N-function. For an N-function Φ and an open set Ω ⊆ RN, the
Orlicz space L (Ω) is defined (see [1]). When Φ satisfies ∆ -condition, that is,
Φ 2
when there are t ≥ 0 and K > 0 such that Φ(2t) ≤ KΦ(t), for all t ≥ t , the
0 0
space L (Ω) is the vectorial space of the measurable functions u:Ω→R such
Φ
that
Φ(|u|) dx<∞.
ZΩ
The space L (Ω) endowed with Luxemburg norm, that is, the norm given by
Φ
|u|
|u| =inf λ>0: Φ dx≤1 ,
Φ
λ
(cid:26) ZΩ (cid:16) (cid:17) (cid:27)
6
is a Banach space. The complement function of Φ, denoted by Φ, is given by
the Legendre transformation, that is
e
Φ(s)=max{st−Φ(t)} for s≥0.
t 0
≥
e
These Φ and Φ are complementary each other. Involving the functions Φ and
Φ, we have the Young’s inequality given by
e
e st≤Φ(t)+Φ(s).
Using the above inequality, it is possible eto prove the following Ho¨lder type
inequality
uv dx ≤2|u|Φ|v|Φe ∀ u∈LΦ(Ω) and v ∈LΦe(Ω).
(cid:12)ZΩ (cid:12)
(cid:12) (cid:12)
Hereafter,(cid:12)(cid:12)wedenot(cid:12)(cid:12)ebyW01LΦ(Ω)theOrlicz-Sobolevspaceobtainedbythe
completion of C (Ω) with norm
0∞
kuk =|u| +|∇u| .
Φ Φ Φ
When Ω is bounded, there is c>0 such that
|u| ≤c|∇u| .
Φ Φ
In this case, we can consider
kuk =|∇u| .
Φ Φ
Another important function related to function Φ, is the Sobolev conjugate
function Φ of Φ defined by
∗
t Φ 1(s)
Φ 1(t)= − ds, t>0.
−
∗ Z0 s(N+1)/N
The function Φ is very important because it is related to some embedding
involving W1L ∗(Ω).
0 Φ
We say that Ψ increases essentially more slowly than Φ near infinity when
∗
Ψ(kt)
lim =0, for all k >0.
t Φ (t)
→∞ ∗
Let Ω be a smooth bounded domain of RN. If Ψ is any N-function
increasing essentially more slowly than Φ near infinity, then the imbedding
W1L (Ω)֒→L (Ω) exists and is compact∗(see [1]).
0 Φ Ψ
The hypotheses (φ ) − (φ ) implies that Φ, Φ, Φ and Φ satisfy ∆ -
1 2 2
∗ ∗
condition. This condition allows us conclude that:
1) u →0 in L (Ω) if, and only if, Φ(u ) dx→e0. e
n Φ n
ZΩ
7
2) L (Ω) is separable and C (Ω)|.|Φ =L (Ω).
Φ 0∞ Φ
3) LΦ(Ω) is reflexive and its dual is LΦe(Ω)(see [1]).
Under assumptions (φ )−(φ ), some elementary inequalities listed in the
1 2
following lemmas are valid. For the proofs, see [24].
Lemma 2.1 Let ξ (t) = min{tl,tm}, ξ (t) = max{tl,tm}, ξ (t) =
0 1 2
min{tl∗,tm∗}, ξ (t)=max{tl∗,tm∗}, t≥0. Then
3
ξ (kuk )≤ Φ(|∇u|) dx ≤ξ (kuk ),
0 Φ 1 Φ
ZΩ
ξ (|u| )≤ Φ (|u|) dx ≤ξ (|u| )
2 Φ∗ 3 Φ∗
ZΩ ∗
and
Φ (t)≥Φ (1)ξ (t).
2
∗ ∗
Lemma 2.2 Let η0(t)=min{tq0,tq1},η1(t)=max{tq0,tq1},t≥0. Then
η (|u| )≤ B(|u|)dx≤η (|u| )
0 B 1 B
ZΩ
and
B(1)η (t)≤B(t)≤B(1)η (t), t∈R.
0 1
Lemma 2.3 Φ(Φ(s))≤Φ(s), s>0.
s
The next result is a version of Brezis-Lieb’s Lemma [10] for Orlicz-Sobolev
e
spaces and the proof can be found in [26].
Lemma 2.4 Let Ω ⊂ RN open set and Φ : R →[0,∞) an N-function satisfies
∆ −condition. If the complementary function Φ satisfies ∆ −condition, (f ) is
2 2 n
bounded in L (Ω), such that
Φ
e
f (x)→f(x) a.s x∈Ω,
n
then
f ⇀f in L (Ω).
n Φ
Corollary 2.1 The imbedding W1L (Ω)֒→L (Ω) exists and is compact.
0 Φ B
Proof: It is sufficiently to show that B increasing essentially more slowly than
Φ near infinity. Indeed,
∗
B(kt) ≤ B(1)B(kt) =B(1)kq1tq1−l∗, k >0.
Φ (t) ξ (t)
2
∗
Since q <l , we get
1 ∗
B(kt)
lim =0.
t + Φ (t)
→ ∞ ∗
8
3 Technical results on locally Lipschitz
functional and variational framework
Inthissection,forthereader’sconvenience,werecallsomedefinitionsandbasic
resultsonthecriticalpointtheoryoflocallyLipschitz continuousfunctionalsas
developed by Chang [13], Clarke [17, 18] and Grossinho & Tersian [25].
Let X be a real Banach space. A functional J : X → R is locally Lipschitz
continuous, J ∈ Lip (X,R) for short, if given u ∈ X there is an open
loc
neighborhood V :=V ⊂X and some constant K =K >0 such that
u V
|J(v )−J(v )|≤K kv −v k, v ∈V, i=1,2.
2 1 2 1 i
The directional derivative of J at u in the direction of v ∈X is defined by
J(u+h+σv)−I(u+h)
J0(u;v)= limsup .
σ
h 0, σ 0
→ ↓
The generalized gradient of J at u is the set
∂J(u)= µ∈X ;hµ,vi≤J0(u;v), v ∈X .
∗
Since J0(u;0) = 0, ∂J(u)(cid:8)is the subdifferential of J0(u;0). (cid:9)Moreover, J0(u;v)
is the support function of ∂J(u) because
J0(u;v)=max{hξ,vi;ξ ∈∂J(u)}.
The generalized gradient ∂J(u) ⊂ X is convex, non-empty and weak*-
∗
compact, and
mJ(u)=min kµkX∗;µ∈∂J(u) .
Moreover, (cid:8) (cid:9)
∂J(u)= J (u) ,if J ∈C1(X,R).
′
A critical point of J is an eleme(cid:8)nt u0(cid:9)∈ X such that 0 ∈ ∂J(u0) and a critical
valueofJ isarealnumbercsuchthatJ(u )=cforsomecriticalpointu ∈X.
0 0
About variational framework, we say that u∈W1L (Ω) is a weak solution
0 Φ
of the problem (P) if it verifies
M Φ(|∇u|) dx φ(|∇u|)∇u∇v dx− φ(u)uvdx− ρv dx=0,
(cid:18)ZΩ (cid:19)ZΩ ZΩ ZΩ
for all v ∈W01LΦ(Ω) and for some ρ∈LBe(Ω) with
f(u(x))≤ρ(x)≤f(u(x)) a.e in Ω,
and moreover the set {x ∈ Ω;| u |≥ a } has positive measure. Thus, weak
0
solutions of (P) are critical points of the functional
9
J(u)=M Φ(|∇u|)dx − Φ(u)dx− F(u) dx,
(cid:18)ZΩ (cid:19) ZΩ ZΩ
t c
where M(t)= M(s)ds. In order to use variational methods, we first derive
Z0
someresultsrelatedto the Palais-Smalecompactnessconditionforthe problem
c
(P).
We saythat a sequence (u )⊂W1L (Ω) is a Palais-Smalesequence for the
n 0 Φ
locally lipschitz functional J associated of problem (P) if
J(un)→c and mJ(un)→0 in (W01LΦ(Ω))∗, (3.1)
where
c= inf max J(η(t))>0
η Γt [0,1]
∈ ∈
and
Γ:={η ∈C([0,1],X):η(0)=0, I(η(1))<0}.
If (3.1) implies the existence of a subsequence (u )⊂(u ) which converges
nj n
in W1L (Ω), we say that these one functionals satisfies the nonsmooth (PS)
0 Φ c
condition.
Note that J ∈Lip (W1L (Ω),R) and from convex analysis theory, for all
loc 0 Φ
w∈∂J(u),
hw,vi=M Φ(|∇u|) dx φ(|∇u|)∇u∇v dx− φ(u)uvdx−hρ,vi,
(cid:18)ZΩ (cid:19)ZΩ ZΩ
for some ρ∈∂Ψ(u), where Ψ(u)= F(u)dx. We have Ψ∈Lip (L (Ω),R),
loc B
ZΩ
∂Ψ(u)∈Le(Ω).
B
The next resultis a versionfor Orlicz-Sobolev spaces of a well-knownresult
of Chang (see [13], [17] and [20, Lemma 3.3]).
Lemma 3.1 Suppose that M , (f ) and (f ) hold. For each u ∈ L (Ω), if
1 2 3 B
ρ∈∂Ψ(u), then
f(u(x))≤ρ(x)≤f(u(x)) a.e x∈Ω,
and if a >0
0
ρ(x)=0 a.e x∈{x∈Ω;|u(x)|<a }.
0
Proof: Considering u,v ∈L (Ω), from definition
B
Ψ(u+h+tv)−Ψ(u+h)
Ψ0(u;v) = limsup
t
h 0,t 0+
→ →
1
= limsup (F(u+h+tv)−F(u+h))dx.
t
h→0,t→0+ ZΩ
10
We set (h ) ⊂ L (Ω) and (t ) ⊂ R such that h → 0 in L (Ω) and
n B n + n B
t →0+. Thus,
n
F(u+h +t v)−F(u+h )
Ψ0(u;v)=limsup n n n dx. (3.2)
t
n→+∞ ZΩ n
Note that from the Mean Value Theorem, (M ), (f ) and (f ) that,
1 2 3
F(u+h +t v)−F(u+h )
n n n
F (u,v):= ≤cb(|θ (x)|)|θ (x)||v|,
n n n
t
n
where
θ (x)∈ min u+h +t v,u+h ,max u+h +t v,u+h , x∈Ω.
n n n n n n n
(cid:20) (cid:21)
(cid:8) (cid:9) (cid:8) (cid:9)
Using monotonicity of b(t)t we get
|F (u,v)|≤cb(|u+h +t v|)|u+h +t v||v|+cb(|u+h |)|u+h ||v|.
n n n n n n n
On the other hand, by lemma 2.3 we have
B(b(|u+h +t v|)|u+h +t v|)≤CB(|u+h +t v|)≤C B(u)+B(h )+η (t )B(v)
n n n n n n n 1 n
aend (cid:0) (cid:1)
B(b(|u+h +t v|)|u+h +t v|)→B(b(|u|)|u|) a.e in Ω,
n n n n
where B(b(|u+h +t v|)|u+h +t v|≤cB(u)∈L1(Ω).
e n n n n e
By Lebesgue’s Theorem we obtain
e
B(b(|u+h +t v|)|u+h +t v|)dx→ B(b(|u|)|u|)dx.
n n n n
ZΩ ZΩ
From 2.4ewe conclude that e
B(b(|u+h +t v|)|u+h +t v|−b(|u|)|u|)dx→0.
n n n n
ZΩ
Moreover,weith obvious changes, we can prove that
B(b(|u+h |)|u+h |−b(|u|)|u|)dx→0.
n n
ZΩ
Thus, by Fatou’selemma that
limsup F (u,v) dx≤ limsupF (u,v) dx. (3.3)
n n
ZΩ ZΩ
From (3.2) and (3.3) we get
Ψ0(u,v)≤ F0(u,v) dx= max{hξ,vi;ξ ∈∂F(u)}dx.
ZΩ ZΩ