Multiple solutions for fourth order elliptic problems with p(x)-biharmonic operators

Abstract

We study the multiplicity of weak solutions to the following fourth order nonlinear elliptic problem with a $p(x)$-biharmonic operator $\begin{cases}\Delta^2_{p(x)}u+a(x)|u|^{p(x)-2}u=\lambda f(x,u)\quad\text{ in }\Omega,\\ u=\Delta u=0\quad\text{ on }\partial\Omega,\end{cases}$ where $\Omega$ is a smooth bounded domain in $\mathbb{R}^N$, $p\in C(\overline{\Omega})$, $\Delta^2_{p(x)}u=\Delta(|\Delta u|^{p(x)-2}\Delta u)$ is the $p(x)$-biharmonic operator, and $\lambda\gt 0$ is a parameter. We establish sufficient conditions under which there exists a positive number $\lambda^{*}$ such that the above problem has at least two nontrivial weak solutions for each $\lambda\gt\lambda^{*}$. Our analysis mainly relies on variational arguments based on the mountain pass lemma and some recent theory on the generalized Lebesgue-Sobolev spaces $L^{p(x)}(\Omega)$ and $W^{k,p(x)}(\Omega)$.