Mountain pass solution for a p(x)-biharmonic Kirchhoff type equation

Document Type : Research Articles

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Department of Mathematics, Faculty of Mathematical Sciences, Farhangian University , Tehran, Iran

Abstract

In this paper we deal with the existence of weak solution for a $p(x)$-Kirchhoff type problem of the following form

$$
\left\{\begin{array}{ll}
-\left(a-b \int_{\Omega}\frac{1}{p(x)}|\Delta u|^{p(x)}\,dx\right)\Delta(|\Delta u|^{p(x)-2}\Delta u)
=\lambda |u|^{p(x)-2}u+g(x,u) & \text{ in } \Omega,\\
u=\Delta u=0 & \textrm{ on } \partial\Omega.
\end{array}\right.
$$
Using the Mountain Pass Theoem, we establish conditions ensuring the existence result.

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