# Three Solutions for Inequalities Dirichlet Problem Driven byp(x)-Laplacian-Like

Authors: Zhou Qing-Mei; Ge Bin
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

A class of nonlinear elliptic problems driven byp(x)-Laplacian-like with a nonsmooth locally Lipschitz potential was considered. Applying the version of a nonsmooth three-critical-point theorem, existence of three solutions of the problem is proved.


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## Body

## 1. Introduction

Since many free boundary problems and obstacle problems may be reduced to partial differential equations with discontinuous nonlinearities, the existence of multiple solutions of the problems with discontinuous nonlinearities has been widely investigated in recent years. In 1981, Chang [1] extended the variational methods to a class of nondifferentiable functionals and directly applied the variational methods for nondifferentiable functionals to prove some existence theorems for PDE with discontinuous nonlinearities. Soon thereafter, Kourogenis and Papageorgiou [2] extend the nonsmooth critical point theory of Chang [1], by replacing the compactness and the boundary conditions. In [3], by using the Ekeland variational principle and a deformation theorem, Kandilakis et al. obtained the local linking theorem for locally Lipschitz functions. In the celebrated work [4], Ricceri elaborated a Ricceri-type variational principle for Gateaux differentiable functionals. Later, Marano and Motreanu [5] extended Ricceri's result to a large class of nondifferentiable functionals and gave an application to a Neumann-type problem involving the p-Laplacian with discontinuous nonlinearities.In this paper, we consider a nonlinear elliptic problem driven byp(x)-Laplacian-like with a nonsmooth locally Lipschitz potential (hemivariational inequality):
(P)-div((1+|∇u|p(x)1+|∇u|2p(x))|∇u|p(x)-2∇u)∈λ∂F(x,u),a.e.inΩ,u=0,on∂Ω,
where Ω⊂ℝN is a bounded domain with C1-boundary ∂Ω. p∈C(Ω), 2≤N<p-:=infx∈Ωp(x)≤p+:=supx∈Ωp(x)<+∞,  F∈C(Ω¯×ℝ), and F:Ω×ℝ→ℝ is a locally Lipschitz with respect to the second variable. By ∂F(x,u), we denote the generalized subdifferential of the locally Lipschitz function u→F(x,u). Our goal is to establish the same results under different assumptions.The study of differential equations and variational problems with variable exponent has been a new and interesting topic. It arises from nonlinear elasticity theory, electrorheological fluids, and so forth (see [6, 7]). The study on variable exponent problems attracts more and more interest in recent years. Many results have been obtained on this kind of problems, for example, [8–14]. Neumann-type problems involving the p(x)-Laplacian have been studied, for instance, in [15–18].Recently, Rodrigues [19] has considered the existence of nontrivial solution for the Dirichlet problem involving the p(x)-Laplacian-like of the type
(1)-div((1+|∇u|p(x)1+|∇u|2p(x))|∇u|p(x)-2∇u)=λf(x,u),000000000000000000000000000000000000000000a.e.inΩ,u=0,on∂Ω,
where Ω⊂ℝN is a bounded domain with smooth boundary ∂Ω, p∈C(Ω¯) with p(x)>2, for all x∈Ω, and f:Ω×ℝ→ℝ satisfies the Caratheodory condition. We emphasize that, in our approach, no continuity hypothesis will be required for the function f with respect to the second argument. So, (P) need not have a solution. To avoid this situation, we consider such function f(x,·) which is locally essentially bounded and fill the discontinuity gap of f(x,·), replacing f by the interval [f1,f2], where
(2)f1(x,t)≔lims→0+essinf|s-t|<δf(x,s),f2(x,t):=lims→0+esssup|s-t|<δf(x,s).
On the other hand, it is well known that if F(x,u)=∫0u‍f(x,t)dt, then F become locally Lipschitz and ∂F(x,u)=[f1(x,u),f2(x,u)] (see [1, 20]).The aim of the present paper is to establish a three-solution theorem for the nonlinear elliptic problem driven byp(x)-Laplacian-like with nonsmooth potential (see Theorem 6) by using a consequence (see Theorem 4) of the three-critical-point theorem established firstly by Marano and Motreanu in [20], which is a non-smooth version of Ricceri's three-critical-point theorem (see [21]). The paper is organized as follows. In Section 2, we present some necessary preliminary knowledge on variable exponent Sobolev spaces and the generalized gradient of the locally Lipschitz function. In Section 3, we give the main result of this paper and use the non-smooth three-critical-point theorem to prove it.


## 2. Preliminary

In order to discuss problem(P), we need some theories on W01,p(x)(Ω) and the generalized gradient of the locally Lipschitz function. Firstly we state some basic properties of space W01,p(x)(Ω) which will be used later (for details, see [10–12]). Denote by S(Ω) the set of all measurable real functions defined on Ω. Two functions in S(Ω) are considered as the same element of S(Ω) when they are equal almost everywhere.PutC+(Ω¯)={p∈C(Ω¯):p(x)>1,∀x∈Ω¯}.Ifp∈C(Ω¯), then write
(3)Lp(x)(Ω)={u∈S(Ω):∫Ω‍|u(x)|p(x)dx<+∞},
with the norm |u|Lp(x)(Ω)=|u|p(x)=inf{λ>0:∫Ω‍|u(x)/λ|p(x)dx≤1}, and
(4)W1,p(x)(Ω)={u∈Lp(x)(Ω):|∇u|∈Lp(x)(Ω)},
with the norm ∥u∥W1,p(x)(Ω)=|u|Lp(x)(Ω)+|∇u|Lp(x)(Ω). Denote by W01,p(x)(Ω) the closure of C0∞(Ω) in W1,p(x)(Ω).We remember that the variable exponent Lebesgue spaces are separable and reflexive Banach spaces. Denote byLq(x)(Ω) the conjugate Lebesgue space of Lp(x)(Ω) with 1/p(x)+1/q(x)=1; then the Hölder-type inequality
(5)∫Ω‍|uv|dx≤(1p-+1q-)|u|p(x)|v|q(x),111111111u∈Lp(x)(Ω),v∈Lq(x)(Ω)
holds. Furthermore, if we define the mapping ρ:Lp(x)(Ω)→ℝ by
(6)ρ(u)=∫Ω‍|u(x)|p(x)dx,
then the following relations hold:
(7)|u|p(x)>1⟹|u|p(x)p-≤ρ(u)≤|u|p(x)p+,|u|p(x)<1⟹|u|p(x)p+≤ρ(u)≤|u|p(x)p-.Proposition 1 (see [12]).
InW01,p(x)(Ω) Poincare's inequality holds; that is, there exists a positive constant C0 such that
(8)|u|p(x)≤C0|∇u|p(x),∀u∈W01,p(x)(Ω).
So |∇u|p(x) is an equivalent norm in W01,p(x)(Ω).We will use the equivalent norm in the following discussion and write∥u∥=|∇u|p(x) for simplicity.Proposition 2 (see [10]).
Ifq∈C+(Ω¯) and q(x)<p*(x) for any x∈Ω¯, then the embedding from W1,p(x)(Ω) to Lq(x)(Ω) is compact and continuous.
Consider the following function:(9)J(u)=∫Ω1p(x)(|∇u|p(x)+1+|∇u|2p(x))dx,00000000000000000000000100u∈W01,p(x)(Ω).
We know that (see [1]).
If one denotesA=J′:W01,p(x)(Ω)→(W01,p(x)(Ω))*, then
(10)〈A(u),v〉=∫Ω(|∇u|p(x)-2+|∇u|2p(x)-21+|∇u|2p(x))(∇u,∇v)ℝNdx,
for all u,v∈W01,p(x)(Ω).Proposition 3 (see [19]).
SetX=W01,p(x)(Ω); A is as shown, then(1)
A:X→X*is a convex, bounded previously; and strictly monotone operator;(2)
A:X→X*is a mapping of type(S)+; that is,un→wuinXandlimsupn→∞〈A(un),un-u〉≤0impliesun→uinX;(3)
A:X→X*is a homeomorphism.
Let(X,∥·∥) be a real Banach space, and let X* be its topological dual. A function f:X→ℝ is called locally Lipschitz if each point u∈X possesses a neighborhood Ωu such that |f(u1)-f(u2)|≤L∥u1-u2∥ for all u1,u2∈Ωu, for a positive constant L depending on Ωu. The generalized directional derivative of f at the point u∈X in the direction h∈X is
(11)f0(u;h)=limsupv→u;t↓0f(v+th)-f(v)t.
The generalized gradient off at u∈X is defined by
(12)∂f(u)={u*∈X*:〈u*,h〉≤f0(u;h)∀h∈X},
which is a nonempty, convex, and w*-compact subset of X, where 〈·,·〉 is the duality pairing between X* and X. One says that u∈X is a critical point of f if 0∈∂f(u).For further details, we refer the reader to the work of Chang [1].Finally, for proving our results in the next section, we introduce the following theorem.Theorem 4 (see [22, 23]).
LetX be a separable and reflexive real Banach space, and let Φ,Ψ:X→ℝ be two locally Lipschitz functions. Assume that there exists u0∈X such that Φ(u0)=Ψ(u0)=0 and Φ(u)≥0 for every u∈X and that there exist u1∈X and r>0 such that(1)
r<Φ(u1);(2)
supΦ(u)<rΨ(u)<r(Ψ(u1)/Φ(u1)), and further, one assumes that function Φ-λΨ is sequentially lower semicontinuous and satisfies the (PS)-condition;(3)
lim∥u∥→∞(Φ(u)-λΨ(u))=+∞ for every λ∈[0,a¯], where
(13)a¯=hrr(Ψ(u1)/Φ(u1))-supΦ(u)<rΨ(u),withh>1.Then, there exist an open interval Λ1⊆[0,a¯] and a positive real number σ such that, for every λ∈Λ1, the function Φ(u)-λΨ(u) admits at least three critical points whose norms are less than σ.


## 3. Existence Results

In this part, we will prove that there exist three solutions for problem(P) under certain conditions.Definition 5.
We say thatI satisfies (PS)c-condition if any sequence {un}⊂W01,p(x)(Ω), such that I(un)→c and m(un)→0, as n→+∞, has a strongly convergent subsequence, where m(un)=inf{∥u*∥X*:u*∈∂I(un)}.
By a solution of(P), we mean a function u∈W01,p(x)(Ω) to which there corresponds a mapping Ω∋x→w(x) with w(x)∈∂F(x,u) for almost every x∈Ω having the property that, for every v∈W01,p(x)(Ω), the function x→w(x)v(x)∈L1(Ω) and
(14)∫Ω‍(|∇u|p(x)-2+|∇u|2p(x)-21+|∇u|2p(x))(∇u,∇v)ℝNdx=λ∫Ω‍w(x)v(x)dx.
We know that W01,p(x)(Ω) is compactly embedded into C(Ω¯) (by N<p-<p*(x)). So there is a constant c0>0 such that |u|∞≤c0∥u∥, for all u∈W01,p(x)(Ω).
SetΦ(u)=∫Ω‍(1/p(x))(|∇u|p(x)+1+|∇u|2p(x))dx,  Ψ(u)=∫Ω‍F(x,u)dx,u∈W01,p(x)(Ω) and φ(u)=Φ(u)-λΨ(u), for all u∈W01,p(x)(Ω).
We know that the critical points ofφ are just the weak solutions of (P).
We consider a non-smooth potential functionF:Ω×ℝ→ℝ such that F(x,0)=0 a.e. on Ω satisfying the following conditions:
H
(
j
): (h1)
F(·,t) is measurable for all t∈ℝ;(h2)
F(x,·) is locally Lipschitz for a.e. x∈Ω;(h3)
there exista∈L∞(Ω)+, c>0 such that
(15)|w|≤a(x)+c|t|α(x)-1,a.e.x∈Ω,∀t∈ℝ,
where w∈∂F(x,t) and 1<α-≤α+<p-;(h4)
there existsq∈C(Ω¯) with p+<q-≤q(x)<p*(x), such that lim|t|→0(F(x,t)/|t|q(x))=0 uniformly a.e. x∈Ω;(h5)
supt∈ℝF(x,t)>0, for all x∈Ω¯.Theorem 6.
Let(h1)–(h5) hold. Then, there are an open interval Λ⊆[0,+∞) and a number σ such that, for every λ belonging to Λ, problem (P) possesses at least three solutions in W01,p(x)(Ω) whose norms are less than σ.Proof.
We observe thatΨ(u) is Lipschitz on Lα(x)(Ω) and, taking into account that α(x)<p*(x), Ψ is also locally Lipschitz on W01,p(x)(Ω) (see Proposition 2.2 of [15]). Moreover it results in ∂Ψ(u)⊆∫Ω‍∂F(x,u)dx (see [24]). The interpretation of ∂Ψ(u)⊆∫Ω‍∂F(x,u)dx is as follows: to every w∈∂Ψ(u) there corresponds a mapping w(x)∈∂F(x,u) for almost all x∈Ω having the property that for every v∈W01,p(x)(Ω) the function w(x)v(x)∈L1(Ω) and 〈w,v〉=∫Ω‍w(x)v(x)dx (see [24]). The proof is divided into the following five steps.
Step  1. We show that φ is coercive.
By(h2), for almost all x∈Ω, t↦F(x,t) is differentiable almost everywhere on ℝ and we have
(16)ddtF(x,t)∈∂F(x,t).
From (h3), there exist positive constants a1,a2 such that
(17)F(x,t)=F(x,0)+∫0t‍ddsF(x,s)ds≤a(x)t+cα(x)|t|α(x)≤a1+a2|t|α(x)
for a.e. x∈Ω and t∈ℝ.
Note that1<α(x)≤α+<p-<p*(x); then by Proposition 2, we have W01,p(x)(Ω)↪Lα(x)(Ω) (compact embedding). Furthermore, there exists c1 such that |u|α(x)≤c1∥u∥.
So, for|u|α(x)>1 and ∥u∥>1, we have ∫Ω‍|u|α(x)dx≤|u|α(x)α+≤c1α+∥u∥α+.
Hence,(18)φ(u)=∫Ω‍1p(x)(|∇u|p(x)+1+|∇u|2p(x))dx-λ×∫Ω‍F(x,u)dx≥2p+∫Ω‍|∇u|p(x)dx-λ∫Ω‍F(x,u)dx≥2p+∥u∥p--λa1meas(Ω)-λa2c1α+∥u∥α+→+∞,
as ∥u∥→+∞.
Step  2. We show that φ is weakly lower semicontinuous.
Letun⇀u weakly in W01,p(x)(Ω), and by Proposition 2, we obtain the following results:
(19)W01,p(x)(Ω)↪Lp(x)(Ω);un→uinLp(x)(Ω);un→ufora.a.x∈Ω;F(x,un(x))→F(x,u(x))fora.a.x∈Ω.
By Fatou's lemma, we have
(20)limsupn→∞∫Ω‍F(x,un(x))dx≤∫Ω‍F(x,u(x))dx.
Thus,
(21)liminfn→∞φ(un)=∫Ω‍1p(x)(|∇un|p(x)+1+|∇un|2p(x))dx-λlimsupn→∞∫Ω‍F(x,un)dx≥∫Ω‍1p(x)(|∇u|p(x)+1+|∇u|2p(x))dx-λ∫Ω‍F(x,u)dx=φ(u).
Step  3.We show that (PS)-condition holds.
Suppose{un}n≥1⊆W01,p(x)(Ω) such that |φ(un)|≤c and m(un)→0 as n→+∞. If un*∈∂φ(un) is such that m(un)=∥un*∥(W01,p(x))*,n≥1, then we know that
(22)un*=Φ′(un)-λwn,
where the nonlinear operator Φ′:W01,p(x)→(W01,p(x))* is defined as
(23)〈Φ′(u),v〉=∫Ω‍(|∇u|p(x)-2+|∇u|2p(x)-21+|∇u|2p(x))(∇u,∇v)ℝNdx,
for all u,v∈W01,p(x)(Ω). From the work of Chang [1], we know that if wn∈∂Ψ(un), then wn∈Lα′(x)(Ω), where 1/α′(x)+1/α(x)=1.
Sinceφ is coercive, {un}n≥1 is bounded in W01,p(x)(Ω) and there exists u∈W01,p(x)(Ω) such that a subsequence of {un}n≥1, which is still denoted as {un}n≥1, satisfies un⇀u weakly in W01,p(x)(Ω). Next we will prove that un→u in W01,p(x)(Ω).
ByW01,p(x)(Ω)→Lα(x)(Ω), we have un→u in Lα(x)(Ω). Moreover, since ∥un*∥*→0, we get |〈un*,un〉|≤εn.
Sinceun*=Φ′(un)-λwn, we obtain
(24)〈Φ′(un),un-u〉-λ∫Ω‍wn(un-u)dx≤εn,∀n≥1.
Moreover, since un→u in Lα(x)(Ω) and {wn}n≥1 are bounded in Lα′(x)(Ω), where 1/α(x)+1/α′(x)=1, one has ∫Ω‍wn(un-u)dx→0. Therefore,
(25)limsupn→∞〈Φ′(un),un-u〉≤0.
But we know that Φ′ is a mapping of type (S+) (by Proposition 3). Thus we obtain
(26)un→uinW01,p(x)(Ω).
Step  4. There exists a u1∈W01,p(x)(Ω)∖{0} such that Ψ(u1)>0.
By(h5), for each x∈Ω¯, there is tx∈ℝ such that F(x,tx)>0.
Forx∈ℝN, denote by Nx a neighborhood of x which is the product of N compact intervals. From (h5) and F(x,t)∈C(Ω¯×ℝ), for any x0∈Ω¯, there are Nx0⊂ℝN, tx0∈ℝ and δ0>0, such that F(x,tx0)>δ0>0 for all x∈Nx0⋂‍Ω¯.
SinceΩ⊆ℝN is bounded, Ω¯ is compact. Then we can find Nx1,Nx2,…,Nxn such that Ω⊂⋃i=1nNxi and Nxi⋂‍Nxj=∂Nxi⋂‍∂Nxj,(i≠j) and, also, we can find tx1,tx2,…,txn∈ℝ, and n positive numbers δ1,δ2,…,δn such that
(27)F(x,txi)>δi>0uniformlyforx∈Nxi⋂‍Ω¯,222222222222222222211111222i=1,2,…,n.
Now, setδ0=min{δ1,δ2,…,δn}, and t0=max{tx1,tx2,…,txn}, and
(28)sup|t|<|t0|;x∈Ω¯|F(x,t)|=M.
Then, we can find a closed set Ωxi⊂int(Nxi⋂‍Ω) such that
(29)meas(Ωxi)>Mmeas(Nxi⋂‍Ω¯)δ0+M,
where meas(A) denote the Lebesgue measure of set A. We consider a function u1∈W01,p(x)(Ω) such that |u1(x)|∈[0,t0] and u1(x)≡txi for all x∈Ωxi. For instance, we can set u1(x)=∑i=1nu1i(x), where u1i∈C0∞(Nxi⋂‍Ω) and
(30)u1i(x)={txi,x∈Ωxi,0≤u1i(x)<txi,x∈(Nxi∩Ω)∖Ωxi.
Then, from (27)–(29), we have
(31)Ψ(u1)=∫Ω‍F(x,u1)dx=∫⋃i=1nNxi∩Ω‍F(x,u1)dx=∫⋃i=1nΩxi‍F(x,u1)dx+∫(⋃i=1nNxi∩Ω)∖⋃i=1nΩxi‍F(x,u1)dx≥∑i=1n‍δimeas(Ωxi)-∑i=1n‍M[meas(Nxi⋂‍Ω)-meas(Ωxi)]>∑i=1n‍[(δ0+M)meas(Ωxi)-Mmeas(Nxi⋂‍Ω¯)]>0.
Step  5. We show that Φ, Ψ satisfy conditions (1) and (2) of Theorem 4.
Letu0=0; then we can easily find Φ(u0)=Ψ(u0)=0.
From (7) and Proposition 1, we have the following:
if∥u∥≥1, then
(32)2p+∥u∥p-≤Φ(u)≤2+|Ω|p-∥u∥p+;
if ∥u∥<1, then
(33)2p+∥u∥p+≤Φ(u)≤2+|Ω|p-.
From (h4), there exist η∈]0,1[ and C3>0 such that
(34)F(x,t)≤C3|t|q(x)≤C3|t|q-,∀t∈[-η,η],x∈Ω.
In view of (h3), if we put
(35)C4=max{C3,supη≤|t|<1a1+a2|t|α-|t|q-,sup|t|≥1a1+a2|t|α+|t|q-},
then we have
(36)F(x,t)≤C4|t|q-,∀t∈ℝ,x∈Ω.
Fix r such that 0<r<1. And when (2/p+)max{∥u∥p-,∥u∥p+}<r<1, by Sobolev Embedding Theorem (W01,p(x)(Ω)↪Lq-(Ω)), we have (for suitable positive constants C5,C6)
(37)Ψ(u)=∫Ω‍F(x,u)dx≤C4∫Ω‍|u|q-dx≤C5∥u∥q-<C6rq-/p-(orC6rq-/p+).
Since q->p+≥p-, we have
(38)limr→0+sup(2/p+)max{∥u∥p-,∥u∥p+}<rΨ(u)r=0.
And so, taking into account (32) and (33),
(39)limr→0+supΦ(u)<rΨ(u)r=0.
From Step 4, there exists u1∈W01,p(x)(Ω)∖{0} such that Ψ(u1)>0. Thanks to (32) and (33), we have
(40)0<2p+max{∥u1∥p-,∥u1∥p+}≤Φ(u1),
and so
(41)Ψ(u1)Φ(u1)>0.
By (32), (33), and (39), there exists r0<(2/p+)max{∥u1∥p-,∥u1∥p+}≤Φ(u1) such that, for each r∈]0,r0[,
(42)supΦ(u)<rΨ(u)<rΨ(u1)Φ(u1).
By choosing r∈]0,r0[, conditions (1) and (2) requested in Theorem 4 are verified and so the proof is complete.



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