Multiplicity results for p(x)-biharmonic equations with nonlinear boundary conditions

Abstract

In this paper, we are interested in the existence of multiple weak solutions of the following fourth-order nonlinear elliptic problem with a $p(x)$-biharmonic operator $\{\begin{array}{ll}{\mathrm{\Delta }}_{p(x)}^{2}u=\lambda f(x)|u{|}^{q(x)-2}u,& x\in \mathrm{\Omega },\\ \frac{\mathrm{\partial }(|\mathrm{\Delta }u{|}^{p(x)-2}\mathrm{\Delta }u)}{\mathrm{\partial }n}=g(x)|u{|}^{r(x)-2}u,& x\in \mathrm{\partial }\mathrm{\Omega },\end{array}$ where $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a bounded domain, ${\mathrm{\Delta }}_{p(x)}^{2}u=\mathrm{\Delta }(|\mathrm{\Delta }u{|}^{p(x)-2}\mathrm{\Delta }u)$ is the operator of fourth order called the $p(x)$-biharmonic operator, $p(x),q(x),r(x)\in C(\overline{\mathrm{\Omega }}),$ and $f\in C(\overline{\mathrm{\Omega }})$, $g\in C(\mathrm{\partial }\mathrm{\Omega })$ are non-negative weight functions with compact support in Ω. Our analysis mainly relies on variational arguments and some recent theory on the generalized Lebesgue–Sobolev spaces $L^{p(x)}(\mathrm{\Omega })$ and $W^{m,p(x)}(\mathrm{\Omega })$.

Keywords:

• $p(x)$-biharmonic operator
• variational methods
• nonlinear boundary conditions

AMS Subject Classifications:

• 35J30
• 35J60
• 31A30

Disclosure statement

No potential conflict of interest was reported by the authors.