For a class of p(x)-biharmonic operators with weights

Abstract

We study the fourth order nonlinear problem with a p(x)-biharmonic operators

$$\begin{aligned} \left\{ \begin{array}{lll} \Delta (|\Delta u|^{p_1(x)-2} \Delta u)+\Delta (|\Delta u|^{p_2(x)-2} \Delta u)&{}=\lambda V_1(x)|u|^{q(x)-2}u-\mu V_2(x)|u|^{\alpha (x)-2}u, \quad &{}x\in \Omega \\ u=\Delta u=0,&{}\quad &{} x\in \partial \Omega \\ \end{array}\right. \end{aligned}$$

where $\Omega \in \mathbb {R}^{N}$ with $N\ge 2$ is a bounded domain with smooth boundary, $\lambda$, $\mu$ are positive real numbers, $p_{1}$, $p_{2}$, q and $\alpha$ are continuous functions on $\overline{\Omega }$, $V_1$ and $V_2$ are weight functions in a generalized Lebesgue spaces $L^{s_1(x)}(\Omega )$ and $L^{s_2(x)}(\Omega )$ respectively such that $V_1$ may change sign in $\Omega$ and $V_2\ge 0$ on $\Omega$. We established an existence results using variational approaches and Ekeland’s variational principle.