MULTIPLICITY RESULTS FOR A CLASS OF p(x)-KIRCHHOFF TYPE EQUATIONS WITH COMBINED NONLINEARITIES

Using the mountain pass theorem combined with the Ekeland variational prin- ciple, we obtain at least two distinct, non-trivial weak solutions for a class of p(x)-Kirchhoff type equations with combined nonlinearities. We also show that the similar results can be obtained in the case when the domain has cylindrical symmetry.


Introduction
Let Ω ⊂ R N , N ≥ 3, be a bounded regular domain.In this paper, we are interested in the multiplicity of solutions for the p(x)-Kirchhoff type equation where M : R + → R is a continuous function, p ∈ C + (Ω) and f : Ω × R → R is a Carathéodory function, satisfying some certain conditions, λ is a parameter.
Since the first equation in (1.1) contains an integral over Ω, it is no longer a pointwise identity; therefore it is often called nonlocal problem.Problem (1.1) is related to the stationary version of the Kirchhoff equation presented by Kirchhoff in 1883, see [15].This equation is an extension of the classical D'Alembert's wave equation by considering the effects of the changes in the length of the string during the vibrations.The parameters in (1.2) have the following meanings: L is the length of the string, h is the area of the cross-section, E is the Young modulus of the material, ρ is the mass density, and P 0 is the initial tension.
In recent years, elliptic problems involving p-Kirchhoff type operators have been studied in many papers, we refer to [2,3,4,17,18,21,22], in which the authors have used different methods to get the existence of solutions for (1.1) in the case when p(x) = p is a constant.
If p : Ω → R is a continuous function, problem (1.1) has been firstly studied by variational methods in [7,8].The p(x)-Laplacian possesses more complicated nonlinearities than Key words and phrases.p(x)-Kirchhoff type operator, Multiple solutions, Mountain pass theorem, Ekeland's principle.
EJQTDE, 2012 No. 42, p. 1 p-Laplacian, for example it is not homogeneous.The study of differential equations and variational problems involving p(x)-growth conditions is a consequence of their applications.Materials requiring such more advanced theory have been studied experimentally since the middle of the last century.In [7], the authors studied problem (1.1) in the special case M (t) = a + bt.
By means of a direct variational approach and the theory of the variable exponent Sobolev spaces, they established in [7] the existence of infinitely many distinct positive solutions whose W 1,p(x) (Ω)-norms and L ∞ -norms tend to zero under suitable hypotheses about nonlinearity.
In [8], the authors considered the problem in the case when M : R + → R is a continuous and non-descreasing function, satisfying the following conditions: Regarding the nonlinearity, they required f to verify the Ambrosetti-Rabinowitz type condition, i.e., there exist µ > p + 1−θ and T > 0 such that 0 < µF (x, t) ≤ f (x, t)t for all t ≥ T and a.e.x ∈ Ω. ( Using the mountain pass theorem in [1], the authors obtained at least one weak solution for (1.1).Also in [8], the authors considered the case when f verifies the condition where p + < γ − ≤ γ + < p − 1−θ .Using the fountain theorem and the dual fountain theorem, the authors obtained a sequence of weak solutions {±u m } with negative energy.In [5,6], the authors studied the multiplicity of solutions for problem (1.1) using the condition (M ′ 1 ) and the three critical points theorem by B. Ricceri.
Motivated by the papers [7,8] and the ideas introduced in [10], the goal of this paper is to study the multiplicity of weak solutions for problem (1.1) with combined nonlinearities.
More exactly, we assume that M : R + → R + is a continuous function satisfying the following conditions: Using the mountain pass theorem and the Ekeland variational principle, we prove that problem (1.1) has at least two distinct, non-trivial weak solutions under suitable conditions on the nonlinear term f .It should be noticed that we do not require the condition (M ′ 1 ) as in [5,6,7,8], for example (M ′ 1 ) is not satisfied when M (t) = t δ−1 for δ > 1, t > 0. We also show that the similar result can be established in the case when the domain Ω has cylindrical symmetry.This comes from the ideas introduced by W. Wang [23], and developed by J. Gao et al. [13,14].Due to the special structure of the domain, we can get the solutions of problem (1.1) with critical and supercritical gowth.In this situation, problem (1.1) is called a Hénon type problem.
In this paper, we consider the problem (1.1) in the particular case with δ 1 , δ 2 are given by the hypothesis (M 1 ) and the following conditions hold: The first result of this paper can be described as follows.
Next, we consider the domain (C 2 ) l c > 0, where More precisely, we consider the following p(x)-Kirchhoff type problem where the function c verifies the conditions (C 1 ) and (C 2 ), and p, a, α, γ : Ω → R are continuous functions, p, γ ∈ S(Ω) := {u : Ω → R : u is real measurable function and u( with δ 1 , δ 2 are given by the hypothesis (M 1 ) and τ is a positive real number defined by Proposition 2.4.
Due to the cylindrical symmetry of the domain Ω, we can deal with problem (1.1) in the supercritical case.To this purpose, we introduce the following space which is a closed subspace of W 1,p(x) 0 (Ω).
Definition 1.3.We say that u ∈ X s is a weak solution of problem (1.7) if and only if EJQTDE, 2012 No. 42, p. 4 Our result concerning problem (1.7) can be described as follows.

Preliminaries
We recall in what follows some definitions and basic properties of the generalized Lebesgue-Sobolev spaces L p(x) (Ω) and W 1,p(x) (Ω) where Ω is an open subset of R N .In that context, we refer to the book of Musielak [20] and the papers of Kováčik and Rákosník [16] and Fan et al. [11,12].Set For any h ∈ C + (Ω) we define For any p(x) ∈ C + (Ω), we define the variable exponent Lebesgue space We recall the following so-called Luxemburg norm on this space defined by the formula dx ≤ 1 .
Variable exponent Lebesgue spaces resemble classical Lebesgue spaces in many respects: they are Banach spaces, the Hölder inequality holds, they are reflexive if and only if 1 < p − ≤ p + < ∞ and continuous functions are dense if p + < ∞.The inclusion between Lebesgue spaces also generalizes naturally: if 0 < |Ω| < ∞ and p 1 , p 2 are variable exponents so that p 1 (x) ≤ p 2 (x) a.e.x ∈ Ω then there exists the continuous embedding L p 2 (x) (Ω) ֒→ L p 1 (x) (Ω).We denote by holds true.
The space L p(x) σ(x) (Ω) endowed with the above norm is a Banach space which has similar properties with the usual variable exponent Lebesgue spaces.In [13], the authors proved the following result which helps us in proving Theorem 1.4.EJQTDE, 2012 No. 42, p. 6 Proposition 2.4 (see [13,Theorem 4 is a bounded regular domain, and Ω 2 ⊂ R k , (k ≥ 2) is a ball of radius R, centered at the origin.Assume that p, γ : Ω are continuous functions, p(x) < N for all x ∈ Ω, p, γ ∈ S(Ω), the function c : Ω → R satisfies the conditions (C 1 ) and (C 2 ).Then, there exists a positive constant τ = τ (c, p, m, k) such that the embedding X s = W 1,p(x) 0,s (Ω) into L γ(x) (Ω, c(x)) is compact and continuous with p(x) < γ(x) < p * (x) + τ for all x ∈ Ω.

Proofs of the main results
Problems (1.6) and (1.7) will be studied using variational methods.We first prove Theorem 1.2 in details, the proof of Theorem 1.4 is similar.Let us associate with problem (1.6) the functional energy J 1 : where where M (t) = t 0 M (s)ds.The functional J 1 associated with problem (1.6) is well defined and of C 1 class on X.Moreover, we have Thus, weak solutions of problem (1.1) are exactly the ciritical points of the functional J. Due to the conditions (M 1 ) and (1.5), we can show that J 1 is weakly lower semi-continuous in X.The following lemma plays an important role in our arguments.
Proof.Let {u m } ⊂ X be a sequence such that where X * is the dual space of X.
We first prove that {u m } is bounded in X.Indeed, we assume the contrary.Then, passing eventually to a subsequence, still denoted by {u m }, we may assume that u m → ∞ as m → ∞.
Thus we may consider that u m > 1 for any m.Using (M 1 ), (M 2 ) we deduce from (3.6) that Since α − < α + < δ 1 p − and β − > δ 2 p + > p + , the sequence {u m } is bounded in X.Thus, there exists u ∈ X such that passing to a subsequence, still denoted by {u m }, it converges weakly to u in X.Then { u m − u } is bounded.By (1.5) and the conditions (A) and (B), the embeddings from X to the weighted spaces L α(x) (Ω, a(x)) and L β(x) (Ω, b(x)) are compact.
Then, using the Hölder inequalities, Propositions 2.1-2.3,we have Since {u m } is bounded in X, passing to a subsequence, if necessary, we may assume that If t 0 = 0 then {u m } converges strongly to u = 0 in X and the proof is finished.If t 0 > 0 then since the function M is continuous, we get Thus, by (M 1 ), for sufficiently large m, we have Thus, {u m } converges strongly to u in X and the functional J 1 satisfies the Palais-Smale condition.
Proof Theorem 1.2.By Lemmas 3.1 and 3.2, all assumptions of the mountain pass theorem in [1] are satisfied.Then we deduce u 1 as a non-trivial critical point of the functional J with J 1 (u 1 ) = c and thus a non-trivial weak solution of problem (1.6).
We now prove that there exists a second weak solution u 2 ∈ X such that u 2 = u 1 .Indeed, by (3.5), the functional J is bounded from below on the unit ball B 1 (0).
Applying the Ekeland variational principle in [9] to the functional J 1 : B 1 (0) → R, it follows that there exists u ǫ ∈ B 1 (0) such that J 1 (u) ≥ r > 0 and inf Then, J 1 (u ǫ ) < inf u∈∂B 1 (0) J 1 (u) and thus, u ǫ ∈ B 1 (0).Now, we define the functional u ǫ is a minimum point of I 1 and thus for all t > 0 small enough and all v ∈ B 1 (0).The above information shows that Letting t → 0 , we deduce that It should be noticed that −v also belongs to B 1 (0), so replacing v by −v, we get which helps us to deduce that J ′ 1 (u ǫ ) X * ≤ ǫ.
Finally, we point out the fact that u 1 = u 2 since J 1 (u 1 ) = c > 0 > c = J 1 (u 2 ).The proof of for all u, v ∈ X s .Thus, weak solutions of problem (1.1) are exactly the ciritical points of the functional J 2 .
From the proof of Theorem 1.2 and Proposition 2.4, using the mountain pass theorem combined with the Ekeland variational principle, we can prove that problem (1.7) has at least two distinct non-trivial weak solutions in X s .