Existence and multiplicity of solutions for a new p ( x )- Kirchho ﬀ problem with variable exponents

: In this article, we study a class of new p ( x )- Kirchho ﬀ problem without satisfying the Ambrosetti - Rabinowitz type growth condition. Under some suitable superliner conditions, we introduce new methods to show the boundedness of Cerami sequences. By using the mountain pass lemma and the symmetric mountain pass lemma, we prove that the p ( x )- Kirchho ﬀ problem has a nontrivial weak solution and in ﬁ nitely many solutions.


Introduction and main results
This work deals with the following nonlocal p(x)-Kirchhoff problem: where ⊂ Ω N is a smooth bounded domain with boundary ∂Ω, a, > b 0 are constants, , and × → f R R : Ω is a continuous function that satisfies some conditions which will be stated later on.
The study of Kirchhoff-type problems with variable exponent growth conditions has received more and more attention because it arises from various applications, we can refer to [1][2][3][4]. We often called problem (1.1) a nonlocal problem because an integral term appears on the left-hand side of the problem (1.1). Problem (1.1) is related to the stationary problem of a model introduced by Kirchhoff [5]. More precisely, Kirchhoff introduced a model given by the following equation: which extends the famous D'Alembert wave equations for free vibration of elastic strings, where ρ P h E , , , 0 , and L are constants that represent some physical meanings.

A
In fact, the (AR) condition guarantees the boundedness of the Palais-Smale sequence of the Euler-Lagrange functional [8,9]. However, it puts strict constraints on the growth of nonlinear term and eliminates many nonlinearities [7,[10][11][12]. Therefore, the authors in [7] give a more weaker superlinear condition: , . It is worth mentioning that the proof of the main results in [6] and [7] actually require that p(x) belong to the modular Poincaré inequality However, the modular Poincaré inequality is not always hold (for details, see [13]). To remove this constraint, we study problem (1.1) without the (AR) condition. Let us assume that f satisfies the following conditions: , for all ∈ x Ω, and a number , f o r a l l , Ω , , , , f o r a l l , Ω ; , , for all ( ) ∈ × x t R , Ω .
Now our main results are as follows: Then there exists Then there exists does not satisfy the (AR) condition. However, it is easy to check that ( ) Throughout this article, the letters C and ( ) = … C i 1, 2, i denote various positive constants. This article is organized as follows. In Section 2, we give some basic properties of the variable exponent Lebesgue space and Sobolev space. In Section 3, we prove the Cerami compactness condition. In Section 4, we prove our main results by the mountain pass lemma and the symmetric mountain pass lemma.

Preliminaries
To discuss problem (1.1), we need some necessary properties on the functional space is defined as follows: is defined as follows: are separable and reflexive Banach spaces [14]. Moreover, there is a constant > C 0, such that is continuous and compact.
, then the following properties hold: , Ω p x 0 1, and the following properties hold: (i) A is convex and sequentially weakly lower semi-continuous; is a strictly monotone operator and homeomorphism.
Definition 2.5. We say that ( ) , it satisfies the following: associated to problem (1.1) is defined as follows: . Hence, the critical points of the functional J λ are weak solutions of problem (1.1).

The Cerami compactness condition
We recall now the definition of the Cerami compactness condition. . Given ∈ c R, we say that J λ satisfies the Cerami condition at any level ∈ c R (( ) C e c condition, for short), if any sequence  Proof. First, we prove that { } u n is bounded in  Since ‖ ‖ > u 1 n , for n sufficiently large, it follows (2.2), (3.1), and Proposition 2.3 that Notice that < It follows (f 4 ) and

(3.5)
By (3.5), (f 2 ), and the Hölder inequality and proposition 2.2, it follows that By proposition 2.4, we can conclude that → u u n as → ∞ n . So J λ satisfies the ( ) C e c condition. □

The proof of main results
To prove Theorems 1.1 and 1.2, we recall the mountain pass theorem and the symmetric mountain pass theorem. (iv) for every finite-dimensional subspace ͠ ⊂ X X, there is ( ) ͠ = > R R X 0 such that ( ) < I u 0 on ͠ ⧹ X B R .
Then the function I has an unbounded sequence of critical values.
Now we prove that function J λ satisfies the mountain pass geometry.