Existence of Weak Solution for p(x)-Kirchhoff Type Problem Involving the p(x)-Laplacian-like Operator by Topological Degree

EL OUAARABI Mohamed,ALLALOU Chakir and MELLIANI Said

Applied Mathematics and Scientific Computing Laboratory,Faculty of Sciences and Techniques,Sultan Moulay Slimane University,Beni Mellal,Morocco.

Abstract. In this paper,we study the existence of”weak solution”for a class of p(x)-Kirchhoff type problem involving the p(x)-Laplacian-like operator depending on two real parameters with Neumann boundary condition.Using a topological degree for a class of demicontinuous operator of generalized (S+) type and the theory of the variable exponent Sobolev space,we establish the existence of”weak solution”of this problem.

Key Words: p(x)-Kirchhoff type problem; p(x)-Laplacian-like operator;weak solution;topological degree methods;variable exponent Sobolev space.

1 Introduction

The study of differential equations and variational problems with nonlinearities and nonstandardp(x)-growth conditions or nonstandard (p(x),q(x))-growth conditions have received a lot of attention.Perhaps the impulse for this comes from the new search field that reflects a new type of physical phenomenon is a class of nonlinear problems with variable exponents (see[1-3]).The motivation for this research comes from the application of similar models in physics to represent the behavior of elasticity[4]and electrorheological fluids (see [5,6]),which have the ability to modify their mechanical properties when exposed to an electric field (see[7-10]),specifically the phenomenon of capillarity,which depends on solid-liquid interfacial characteristics as surface tension,contact angle,and solid surface geometry.

Let Ω be a bounded domain in RN(N＞1) with smooth boundary denoted by∂Ω,R∈L∞(Ω),p(x),(that will be defined in the Preliminaries),and letµandλbe two real parameters.

In this paper,we establish the existence of weak solution for a class ofp(x)-Kirchhoff type problem involving thep(x)-Laplacian-like operator depending on two real parameters with Neumann boundary condition of the following form:

is thep(x)-Laplacian-like operator,g:Ω×R→R andf:Ω×R×RN→R are Carathéodory functions that satisfy the assumption of growth andM:R+→R+is a continuous function.

Problems related to (1.1) have been studied by many scholars,for example,Ni and Serrin[11,12]considered the following equation

The operator on the left-hand side of (1.2) is most often denoted by the specified mean curvature operator andis the Kirchhoff stress term.

which is called thep(x)-Kirchhoff type problem.In this case,Dai et al.[19],by a direct variational approach,established conditions ensuring the existence and multiplicity of solution to (1.3).Furthermore,the problem (1.3) is a generalization of the stationary problem of a model introduced by Kirchhoff[20]of the following form:

whereρ,ρ0,h,E,Lare all constants,which extends the classical D’Alembert’s wave equation,by considering the effect of the changing in the length of the string during the vibration.

Lapa et al.[21]showed,by using a Fredholm-type result for a couple of nonlinear operator,and the theory of variable exponent Sobolev space,the existence of weak solution for the problem (1.1),under no-flux boundary conditions,in the case whenµ=R=0,λ=1 andfindependent of∇u(see also[22-25]).

In the present paper,we will generalize these works,by proving,under a conditions onM,gandf,the existence of a weak solution for the problem (1.1).Note that the problem (1.1) does not have a variational structure,so the most usual variational methods can not used to study it.To attack it we will employ a topological degree for a class of demicontinuous operator of generalized (S+) type of[26].

The remainder of the paper is organized as follows.In Section 2,we review some fundamental preliminaries about the functional framework where we will treat our problem.In Section 3,we introduce some classes of operators of generalized (S+) type,as well as the Berkovits topological degrees.Finaly,in Section 4,we give our basic assumptions,some technical lemmas,and we will state and prove the main result of the paper.

2 Preliminaries

In the analysis of problem (1.1),we will use the theory of the generalized Lebesgue-Sobolev spaceLp(x)(Ω) andW1,p(x)(Ω).For convenience,we only recall some basic facts with will be used later,we refer to[27-32]for more details.

Let Ω be a smooth bounded domain in RN(N＞1),with a Lipschitz boundary denoted by∂Ω.Set

where the infinimum is taken on all possible decompositionsu=u0-divFwithu0∈Lp′(x)(Ω) andF=(u1,...,uN)∈(Lp′(x)(Ω))N.

3 A review on the topological degree theory

Now,we give some results and properties from the theory of topological degree.The readers can find more information about the history of this theory in[26,33].

In what follows,letXbe a real separable reflexive Banach space andX*be its dual space with dual pairing 〈·,·〉 and given a nonempty subset Ω ofX.Strong (weak) convergence is represented by the symbol→(⇀).

Definition 3.1.Let Y be a real Banach space.A operator F:Ω⊂X →Y is said to be:

1.bounded,if it takes any bounded set into a bounded set.

2.demicontinuous,if for any sequence(un)⊂Ω,un →u implies that F(un)⇀F(u).

3.compact,if it is continuous and the image of any bounded set is relatively compact.

Definition 3.2.A mapping F:Ω⊂X →X* is said to be:

1.of class(S+),if for any sequence(un)⊂Ωwith un ⇀u andlimsupn→∞〈Fun,un-u〉≤0,we have un →u.

2.quasimonotone,if for any sequence(un)⊂Ωwith un⇀u,we havelimsupn→∞〈Fun,unu〉≥0.

Definition 3.3.Let T:Ω1⊂X→X*be a bounded operator such thatΩ⊂Ω1.For any operator F:Ω⊂X →X,we say that

1.F of class(S+)T,if for any sequence(un)⊂Ωwith un⇀u,yn:=Tun⇀y andlimsupn→∞〈Fun,yn-y〉≤0,we have un →u.

2.F has the property(QM)T,if for any sequence(un)⊂Ωwith un ⇀u,yn:=Tun ⇀y,we havelimsupn→∞〈Fun,y-yn〉≥0.

In the sequel,for anyT ∈F1(Ω),we consider the following classes of operators:

Now,letObe the collection of all bounded open sets inXand we define

Lemma 3.1.([33,Lemma 2.3])Letbe continuous and S :D(S)⊂X*→X be demicontinuous such that,where E is a bounded open set in a real reflexive Banach space X.Then the following statements are true:

1.If S is quasimonotone,then,where I denotes the identity operator.

2.If S is of class(S+),then.

Definition 3.4.Suppose that E is bounded open subset of a real reflexive Banach space X,T ∈is continuous and F,.The affine homotopyΛ:defined by

is called an admissible affine homotopy with the common continuous essential inner map T.

Remark 3.1.([33,Lemma 2.5]) The above affine homotopy is of class (S+)T.

Next,as in[33]we give the topological degree for the classF(X).

Theorem 3.1.Letthen,there exists a uniquedegree function d:M→Zthat satisfies the following properties:

1.(Normalization) For any h∈E,we have

2.(Additivity) Let.If E1and E2are two disjoint open subsets of E such that,then we have

3.(Homotopy invariance) IfΛ:is a bounded admissible affine homotopy with a common continuous essential inner map and h:[0,1]→X is a continuous path in X such that h(t)Λ(t,∂E)for all t∈[0,1],then

4.(Existence) If d(F,E,h)̸=0,then the equation Fu=h has a solution in E.

5.(Boundary dependence) If F,coincide on ∂E and,then

Definition 3.5.([33,Definition 3.3])The above degree is defined as follows:

where dB is the Berkovits degree[26]and E0is any open subset of E with F-1(h)⊂E0and F is bounded on.

4 Existence of weak solution

In this section,we will discuss the existence of weak solution of (1.1).

We assume that Ω⊂RN(N＞1) is a bounded domain with a Lipschitz boundary∂Ω,with 1＜c-≤c(x)≤c+＜p-,M:R+→R+,g:Ω×R→R andf:Ω×R×RN →R are functions such that:

(M0)M:[0,+∞)→(m0,+∞) is a continuous and increasing function withm0＞0.

(A1)fis a Carathéodory function.

(A2) There existsϱ＞0 andγ∈Lp′(x)(Ω) such that

(A3)gis a Carathéodory function.

(A4) There areσ＞0 andν∈Lp′(x)(Ω) such that

for a.e.x∈Ω and all (ζ,ξ)∈R×RN,whereq,s∈with 1＜q-≤q(x)≤q+＜p-and 1＜s-≤s(x)≤s+＜p-.

Remark 4.1.We make the following observations:

• Note that,for allu,ϑ∈W1,p(x)(Ω)

is well defined (see[21]).

•R(x)|u|c(x)-2u ∈Lp′(x)(Ω),µg(x,u)∈Lp′(x)(Ω),λ f(x,u,∇u)∈Lp′(x)(Ω) underu ∈W1,p(x)(Ω),the assumptions (A2) and (A4) and the given hypotheses about the exponentsp,c,qandsbecause:γ ∈Lp′(x)(Ω),ν ∈Lp′(x)(Ω),r(x)=(q(x)-1)p′(x)∈withβ(x)＜p(x) andκ(x)=withκ(x)＜p(x).

Then,by Remark 2.2 we can conclude that

Hence,sinceϑ∈Lp(x)(Ω),we have

This implies that,the integral

is finite.

Then,let us introduce the definition of a weak solution for (1.1).

Definition 4.1.We say that a function u ∈W1,p(x)(Ω)is a weak solution of(1.1),if for any ϑ∈W1,p(x)(Ω),it satisfies the following:

Let us now give two lemmas that will be used later.

Lemma 4.1.If(M0)holds,then the operator T:(Ω)→W-1,p′(x)(Ω)defined by

is continuous,bounded,strictly monotone and is of type(S+).

Proof.Let us consider the following functional:

such thatM(τ) satisfies the assumption (M0).

From[21],it is obvious thatJis a continuouslydifferentiable function whosederivative at the point(Ω) is the functionalT(u):=J′(u)∈W-1,p′(x)(Ω) given by

for allu,(Ω) where 〈·,·〉means the duality pairing betweenW-1,p′(x)(Ω) and

By using the similar argument as in [21,Theorem 3.1.] and in [13,Proposition 3.1.],we conclude thatTis continuous,bounded,strictly monotone and is of type (S+).

Lemma 4.2.Assume that the assumptions(A1)-(A4)hold,then the operator

is compact.

Proof.In order to prove this lemma,we proceed in four steps.

Step 1:Let Y:W1,p(x)(Ω)→Lp′(x)(Ω) be an operator defined by

In this step,we prove that the operator Y is bounded and continuous.

First,letu∈W1,p(x)(Ω),bearing (A4) in mind and using (2.5) and (2.6),we infer

Then,we deduce from Remark 2.3 and,that

that means Y is bounded onW1,p(x)(Ω).

Second,we show that the operator Y is continuous.

To this purpose letun→uinW1,p(x)(Ω).We need to show that Yun→YuinLp′(x)(Ω).We will apply the Lebesgue’s theorem.

Note that ifun →uinW1,p(x)(Ω),thenun →uinLp(x)(Ω).Hence there exist a subsequence (uk) of (un) andϕinLp(x)(Ω) such that

for a.e.x∈Ω and allk∈N.

Hence,from (A2) and (4.1),we have

for a.e.x∈Ω and for allk∈N.

On the other hand,thanks to (A3) and (4.1),we get,ask→∞

Seeing that

then,from the Lebesgue’s theorem and the equivalence (2.4),we have

and consequently

that is,Y is continuous.

Step 2:We define the operator Ψ:W1,p(x)(Ω)→Lp′(x)(Ω) by

We will prove that Ψ is bounded and continuous.

It is clear that Ψ is continuous.Next we show that Ψ is bounded.Letu ∈W1,p(x)(Ω) and using (2.5) and (2.6),we obtain

and consequently,Ψ is bounded onW1,p(x)(Ω).

Step 3:Let us define the operator Φ:W1,p(x)(Ω)→Lp′(x)(Ω) by

We will show that Φ is bounded and continuous.

Letu∈W1,p(x)(Ω).According to (A2) and the inequalities (2.5) and (2.6),we obtain

and consequently Φ is bounded onW1,p(x)(Ω).

It remains to show that Φ is continuous.Letun →uinW1,p(x)(Ω),we need to show that Φun →ΦuinLp′(x)(Ω).We will apply the Lebesgue’s theorem.

Note that ifun→uinW1,p(x)(Ω),thenun→uinLp(x)(Ω) and∇un→∇uin (Lp(x)(Ω))N.Hence,there exist a subsequence (uk) andϕinLp(x)(Ω) andψin (Lp(x)(Ω))Nsuch that

for a.e.x∈Ω and allk∈N.

Hence,thanks to (A1) and (4.2),we get,ask→∞

On the other hand,from (A2) and (4.3),we can deduce the estimate

for a.e.x∈Ω and for allk∈N.

Seeing that

and taking into account the equality

then,we conclude from the Lebesgue’s theorem and (2.4) that

and consequently

and then Φ is continuous.

Step 4:LetI*:Lp′(x)(Ω)→W-1,p′(x)(Ω) be the adjoint operator of the operatorI:W1,p(x)(Ω)→Lp(x)(Ω).

We then define

On another side,taking into account thatIis compact,thenI*is compact.Thus,the compositionsI*◦Y,I*◦Ψ andI*◦Φ are compact,that meansS=I*◦Y+I*◦Ψ+I*◦Φ is compact.With this last step the proof of Lemma 4.2 is completed.

We are now in the position to give the existence result of weak solution for (1.1).

Theorem 4.1.Assume that(A1)-(A4)and(M0)hold,then the problem(1.1)admits at least one weak solution u in W1,p(x)(Ω).

Proof.The basic idea of our proof is to reduce the problem (1.1) to a new one governed by a Hammerstein equation,and apply the theory of topological degree introduced in Section 3 to show the existence of a weak solution to the state problem.

For allu,ϑ ∈W1,p(x)(Ω),we define the operatorsTandS,as defined in Lemmas 4.1 and 4.2 respectively,

Consequently,the problem (1.1) is equivalent to the equation

Taking into account that,by Lemma 4.1,the operatorTis a continuous,bounded,strictly monotone and of class (S+),then,by[34,Theorem 26 A],the inverse operator

is also bounded,continuous,strictly monotone and of class (S+).

On another side,according to Lemma 4.2,we have that the operatorSis bounded,continuous and quasimonotone.

Consequently,following Zeidler’s terminology[34],Eq.(4.4) is equivalent to the following abstract Hammerstein equation

Seeing that (4.4) is equivalent to (4.5),then to solve (4.4) it is thus enough to solve (4.5).In order to solve (4.5),we will apply the Berkovits topological degree introduced in Section 3.

Let us set

Next,we show thatRis bounded in∈W-1,p′(x)(Ω).

Let us putu:=Aϑfor allϑ∈R.Taking into account that|Aϑ|1,p(x)=|u|1,p(x),then we have the following two cases:

First case:If|u|1,p(x)≤1,then|Aϑ|1,p(x)≤1,that meansis bounded.

Second case:If|u|1,p(x)＞1,then,we deduce from (2.9),(A2) and (A4),the inequalities (2.7) and (2.6) and the Young’s inequality that

then,according to Remark 2.3,,we get

On the other hand,we have that the operator isSis bounded,thenS◦Aϑis bounded.Thus,thanks to (4.5),we have thatRis bounded inW-1,p′(x)(Ω).

However,∃r＞0 such that

which leads to

whereRr(0) is the ball of center 0 and radiusrinW-1,p′(x)(Ω).

Moreover,by Lemma 3.1,we conclude that

On another side,taking into account thatI,SandAare bounded,thenI+S◦Ais bounded.Hence,we infer that

Next,we define the homotopy

Applying the homotopy invariance property of the degreedseen in Theorem 3.1,we obtain

Then,by the normalization property of the degreed,we haved(I,Rr(0),0)=1 and consequentlyd(I+SoA,Rr(0),0)=1.

Sinced(I+SoA,Rr(0),0)̸=0,then by the existence property of the degreedstated in Theorem 3.1,we conclude that there existsϑ∈Rr(0) which verifies

Finally,we infer thatu=Aϑis a weak solution of (1.1).The proof is completed.