Multiplicity of solutions for a nonlinear nonlocal problem with variable exponent

Abstract

This work deals with a class of value problem involving the $p(x)$-biharmonic elliptic equation \begin{equation*} \begin{gathered} M_1(\int_{\Omega}\frac{1}{p(x)}|\Delta u|^{p(x)}dx)\Delta^2_{p(x)}u-M_2(\int_{\Omega}\frac{1}{p(x)}|\nabla u|^{p(x)}dx)\Delta_{p(x)}u=\lambda f(x,u)+\mu g(x,u) \quad\text{in }\Omega,\\ u=\Delta u=0,\quad x\in \partial \Omega, \end{gathered} \end{equation*} where $\Omega$ is a bounded domain in $\mathbb{R}^N,~N\geq1.$ with smooth boundary $\partial \Omega.$ Our technical method is based on a theorem obtained by B. Ricceri.