This paper is concerned with the existence and multiplicity of weak solutions for a $p(x)$\-Kirchhoff type problem of the following form 

$$
\left\{ \begin{alignedat}{2} M\left(\int_{\Omega}\frac{1}{p(x)}|\Delta u|^{p(x)}\,dx\right)\Delta(|\Delta u|^{p(x)-2}\Delta u) &=f(x,u)& &\text{ in } \Omega \\ u&=\Delta u =0 &\quad &\textrm{ on } \partial\Omega, \end{alignedat}\right.
$$

 by using the mountain pass theorem of Ambrosetti and Rabinowitz and Ekeland's variational principle in two cases when the Carath\\'{e}odory function $f(x,u)$ having special structure.

Existence and Multiplicity of Solutions for Kirchhoff Type Problems Involving $p (x)$ -Biharmonic Operators

Ghasem A. Afrouzi

Maryam Mirzapour

Nguyen Thanh Chung

Abstract

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