Existence of Solutions for Some p ( x )-polyharmonic Elliptic Kirchhoff Equations

In this paper, we study the existence of solution for some p(x)-polyharmonic Kirchhoff equations. The latter is allowed to vanish at the origin (degenerate case). Firstly, we study the existence of solutions of approximate equations. Secondly, we prove the existence of the solutions of the original equation. The main tool is the Schauder’s Theorem.


Introduction
In this paper, we prove the existence of solution of Dirchlet problems involving the p-polyharmonic operators s p ∆ . We consider  (1) where N Ω ⊂  is a bounded domain, 2 p ≥ , 1, 2, s =  , ⋅ is denoted in section 2, and which becomes the usual p-Laplacian for 1 s = . Kratochvl and Necâs introduced the p-biharmonic operator in [1] [2] [3] to study the physical equations, the p-biharmonic operator for 2 s = and the polyharmonic operator for  (3) We introduce for 1, 2, s =  , the main s-order differential operator Note that s  is an n-vectorial operator when s is odd and 1 n > , while it is a scalar operator when s is even.
In our hypothesis, the Kirchhoff function when M is of the type (6) and 0, 0 a b > ≥ , problem (1) is said to be non-degenerate, while it is called degenerate if 0 a = . Besides, problem (2) reduces to the usual well-known quasilinear elliptic equation while 0, The existence of positive solutions of non-degenerate Kirchhoff-type problems has been proved in [4] [5] for 1 L = . The novelty of this paper is to treat the degenerate case with allowing Kirchhoff function to take the zero value. Several authors have considered fourth order problems with nonlinear boundary conditions involving third order derivatives, see [6]. The classical counterpart of our problem models containning several interesting phenomena were deeply studied in physicals even in the one-dimensional case. It dates back to 1883 when Kirchhoff proposed his celebrated equation: and there exists 0 Q > such that, for x ∈ Ω a.e., There is assumption that ( ) The author exploits the symmetric mountain pass theorem to proves the multiplicity of solutions for p(x)-polyharmonic elliptic Kirchhoff equations in [17].
In contrast, in this paper, the keystone of the proofs them is the deduction, by condition (7), (8), of the L ∞ -estimate of the approximated solutions, we prove the problem existing a solution This paper is organized as follows. In Section 2, we introduce some basic no-

Notations and Preliminaries
In this section, we briefly introduce some basic results and notations. Let Ω be a bounded domain in N  , we denote a multi-index ( ) , such that the corresponding partial differentation: By the poncaré inequality, there exists a positive constant where ( ) Ω is a separable, uniformly convex, reflexive, real Banach space.
Note that, when 2 p = , this norm is introduced by the inner product We study problem (1) for a solution, we understand: where s  is the operator in (4) and

Existence and Uniqueness of Solution for (1)
In order to study the solution of problem (1), we consider problems: Let us define: and that we choose 0 0 k > , such that We define: x y R ∈ , and 2 p ≥ , by Hölder's inequality, we obtain Step 2. We prove S is compact, first we take a sequence { } with S is a positive constant, independent of k, such that, Because of the continuity of S, necessarily ( )   (21), next we will prove the problem (1).
We will use the following function defined for t R ∈ , by L Ω convergence of ( ) we pass to the limit in the problem (21), we prove that u satisfies (1), with

A p(x)-polyharmonic Kirchhoff Equation
In this section, we begin by recalling some basic results on the variable exponent Lebesgue and Sobolev spaces, see details in [22] [23].
As before, we define: Let h be the function in ( ) C Ω , an important role in manipulating the generalized Lebesgue-Sobolev spaces is played by Let p be a fixed function in ( ) C + Ω . We endow the Luxemburg norm: p′ be the function obtained by conjugating the exponent p pointwise, so that ( ) ( ) for all x ∈ Ω , the p′ belongs to ( ) Note that, by Hölder-type inequality is valid: From now on we also assume that So, the proof will be divided into two steps.
Step 1: We prove the continuity. In order to do this, we define