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

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{۱}{p(x)}|\Delta u|^{p(x)}\,dx\right)\Delta(|\Delta u|^{p(x)-۲}\Delta u)=\lambda |u|^{p(x)-۲}u+g(x,u) & \text{ in } \Omega,\\u=\Delta u=۰ & \textrm{ on } \partial\Omega.\end{array}\right.Using the Mountain Pass Theoem, we establish conditions ensuring the existence result.

Keywords:

p(x)-biharmonic equation , Kirchhoff type problems , Mountain pass theorem