Existence and multiplicity of solutions for a p ( x )-Kirchhoff type problems

This paper is concerned with the existence and multiplicity of solutions for a class of p(x)-Kirchhoff type equations with Neumann boundary condition. Our technical approach is based on variational methods.


Introduction
In this work, we study the existence and multiplicity of solutions for the nonlocal elliptic problem under Neumann boundary condition: −M (t) div(|∇u| p(x)−2 ∇u) − a(x)|u| p(x)−2 u = λf (x, u) in Ω ∂u ∂ν = 0 on ∂Ω, where Ω is an open bounded subset of R N (N ≥ 2), with smooth boundary, ∂u ∂ν is the outer unit normal derivative, a ∈ L ∞ (Ω), with ess inf Ω a > 0, λ > 0 and p(x) ∈ C + (Ω) with In the statement of problem (1.1), f : Ω × R → R is an Carathéodory function and M (t) is a continuous function with t := Ω 1 p(x) |∇u| p(x) + a(x)|u| p(x) dx.The p(x)-Laplacian operator possesses more complicated nonlinearities than the p-Laplacian operator, mainly due to the fact that it is not homogeneous.The study of various mathematical problems with variable exponent growth condition has been received considerable attention in recent years, we can for example refer to [1,4,17,24,29,35].This great interest may be justified by their various physical applications.In fact, there are applications concerning elastic mechanics [41], electrorheological fluids [38,39], image restoration [13], dielectric breakdown, electrical resistivity and polycrystal plasticity [7,8] and continuum mechanics [5].
As it is well know, problem (1.1) is related to the stationary problem of a model introduced by Kirchhoff [36].More precisely, Kirchhoff introduced a model given by the following equation where ρ, ρ 0 , h, E, L are constants,which extends the classical D'Alembert's wave equation, by considering the effects of the changes in the length of the strings during the vibrations.A distinguishing feature of the which is has attracted much attention after Lions's paper [31], where a functional analysis frame work for the problem was proposed; see, e.g., [6,12,16] for some interesting results.Moreover, nonlocal problems like can be used for modeling several physical and biological systems where u describes a process which depends on the average of it self, such as the population density, see [3].The study of Kirchhoff type equations has already been extended to the case involving the p-Laplacian see, e.g., [11,26,30].In [11], the authors present several sufficient conditions for the existence of positive solutions to a class of nonlocal boundary value problems of the p-Kirchhoff type equation.However, to our knowledge, there is not a great number of papers which have dealt with nonlocal p(x)-Laplacian equations.We refer the reader to [14,18,19,20,34] and the references therein for an overview on this subject.Our aim is to establish the existence and multiplicity results for problem (1.1) through variational methods.First we will exploit a critical point theorem by Bonanno ( [9], Theorem 5.1) which provides for the existence of a local minima Existence and multiplicity of solutions 203 for a parameterized abstract functional, and a classical theorem of Ambrosetti-Rabinowitz, to guarantee that (1.1) has at least two distinct nontrivial weak solutions (Theorem 3.1).Next, we will get the existence of a nontrivial solution of the problem (1.1) where the nonlinearity f (x, u) does not satisfy Ambrosetti-Rabinowitz condition (Theorem 3.2), by employing a local minimum theorem ( [9], Theorem 5.3).These results can be viewed as generalizations to the nonlocal and variable exponent space setting of some results obtained in [10,33].

Preliminaries
Our main tools are two consequences of a local minimum theorem [9, Theorem 3.1] which are recalled below.Given X a set and two functionals Φ, Ψ : X → R, put ) for all r 1 , r 2 ∈ R, with r 1 < r 2 , and for all r ∈ R.
Theorem 2.1 ( [9], Theorem 5.1).Let X be a reflexive real Banach space, Φ : X → R be a sequentially weakly lower semicontinuous, coercive and continuously Gâteaux differentiable function whose Gâteaux derivative admits a continuous inverse on X * , Ψ : X → R be a continuously Gâteaux differentiable function whose Gâteaux derivative is compact.Put I λ = Φ − λΨ and assume that there are r 1 , r 2 ∈ R, with where β and ρ 1 are given by (2.1) and (2.2).Then, for each λ ∈ Let X be a real Banach space; Φ : X → R be a continuously Gâteaux differentiable function whose Gâteaux derivative admits a continuous inverse on X * , Ψ : X → R be a continuously Gâteaux differentiable function whose Gâteaux derivative is compact.Fix inf X Φ < r < sup X Φ and assume that ρ 2 (r) > 0, where ρ 2 is given by (2.3), and for each λ > 1 ρ 2 (r) the function In the sequel, let p(x) ∈ C + (Ω), where The variable exponent Lebesgue space is defined by and the variable exponent Sobolev space is defined by 27,28]).The spaces L p(x) (Ω) and W 1,p(x) (Ω) are separable, uniformly convex, reflexive Banach spaces.The conjugate space of L p(x) (Ω) is L q(x) (Ω), where q(x) is the conjugate function of p(x); i.e., where Existence and multiplicity of solutions

205
Since a ∈ L ∞ (Ω) with ess inf Ω a > 0, we see that .a is a norm on X equivalent to .W 1,p(x) (Ω) .Now, we introduce the modular ρ : X → R defined by for all u ∈ X.Here, we give some relations between the norm ||.|| a and the modular ρ.
Definition 2.6.We say that u ∈ X is a weak solution of problem (1.1) if We introduce the functionals Φ, Ψ : X → R, defined by for all u ∈ X, where It is well known that Φ and Ψ are well defined and continuously Gâteaux differentiable whose Gâteaux derivatives at point u ∈ X are given by for all v ∈ X.
We need the following theorem in the proofs of our main results.(ii) Φ ′ is strictly monotone (iv) Φ ′ admits a continuous inverse on X * .
Existence and multiplicity of solutions 207

Main results
In order to introduce our result, given two positive constants c and d with and k is given by (2.6).
Theorem 3.1.If (f 0 ), (f 1 ), (M 0 ) and (M 1 ) hold, and there exist three constants Then, for each λ ∈ .Proof: Let Φ, Ψ be the functionals defined in (2.7).Since p − > 1, for each u ∈ X such that u a ≥ 1 we have So, Φ is a coercive.From Theorem 2.7, of course, Φ ′ admits a continuous inverse on X * , moreover, Ψ has a compact derivative, it results sequentially weakly continuous.Hence Φ and Ψ satisfy all regularity assumptions requested in Theorem 2.1 and that the critical points of the functional Φ − λΨ in X are exactly the weak solutions of problem (1.1).So, our aim is to verify condition (2.4) of Theorem 2.1.
To this end, let u 0 (x) = d for all x ∈ Ω, and put Then, in virtu of the strict monotonicity of M , we get Hence, it follows from (3.1) that . By (M 0 ) and Proposition 2.5, we obtain This together with (2.6), yields On the other hand, arguing as before we obtain Existence and multiplicity of solutions 209 and So, by our assumption it follows that Hence, from Theorem 2.1 for each λ ∈ . Now we prove the existence of the second local minimum distinct from the first one.To this purpose, we verify the hypotheses of the mountain pass theorem for the functional I λ .Clearly I λ is of class C 1 and I λ (0) = 0.The first part of proof guarantees that u 1 ∈ X is a local nontrivial local minimum for I λ in X. Therefore there is ̺ > 0 such that inf u−u1 a =̺ I λ (u) ≥ I λ (u 1 ), so condition [37, (I 1 ), Theorem 2.2] is verified.From condition (f 1 ), by standard computations, there is a positive constant c 1 such that By integrating (M 1 ), we get Hence, from (3.7) and (3.8), for u ∈ X\{0} and t > 1, we obtain 1−θ .So the condition [37, (I 2 ), Theorem 2.2] is verified.Now, we verify that I λ satisfies the (PS)-condition.To this end, suppose that (u n ) ⊂ X is a (PS)-sequence; i.e., there is M > 0 such that Let us show that (u n ) is bounded in X.Using hypothesis (f 1 ) and (M 1 ), for n large enough, we have Since µ > p + 1−θ , (u n ) is bounded, for a subsequence still denoted (u n ), we can assume that u n ⇀ u in X, then I ′ λ (u n ), u n − u → 0. Thus, we have From (f 0 ) and Proposition 2.3, we get that Ω f (x, u n )(u n − u) dx → 0. there-Existence and multiplicity of solutions 211 fore, one has In view of condition (M 0 ), we obtain We write |∇u| p(x) + a(x)|u| p(x) dx.
Using Theorem 2.7, the mapping J ′ : X → X * is of (S + ) type.Then we have u n → u.Consequently, the classical theorem of Ambrosetti and Rabinowitz ensures a critical point u 2 such that I λ (u 2 ) > I λ (u 1 ).So admits at least two nonnegative weak solutions.
Proof: Clearly, one has F (x, s) = α(x)G(s) for all (x, s) ∈ Ω × R. Therefore, taking into account that G is a nondecreasing function, one has Therefore, Theorem 3.1 ensure the existence of at last two solutions, and by standard argument we see that they are nonnegative.✷ Finally, we give an application of Theorem 2.2.

1 A
d (c1) , 1 A d (c2) , problem (1.1) admits at least two nontrivial weak solutions u 1 and u 2 such that p −