Existence results for perturbed weighted p(x)-biharmonic problem with Navier boundary conditions

Abstract

In this paper, we are interested in the study of the following problem with Navier boundary conditions $\begin{array}{l}\mathrm{\Delta }\left(|x{|}^{p\left(x\right)}|\mathrm{\Delta }u{|}^{p\left(x\right)-2}\mathrm{\Delta }u\right)=\lambda V\left(x\right)|u{|}^{q\left(x\right)-2}u\text{in}\mathrm{\Omega },\\ u=\mathrm{\Delta }u=0\text{on}\mathrm{\partial }\mathrm{\Omega },\end{array}$ where Ω is a smooth bounded domain in ${\mathbb{R}}^n$, $\lambda >0$, the potential V is in some generalized Sobolev space, and $p,q:\mathrm{\Omega }\to [1,\mathrm{\infty })$ are continuous functions. The main tools used here are based on the variational method combined with the Mountain Pass theorem and Ekeland variational principle.

COMMUNICATED BY:

• Y. Xu

Keywords:

• $p\left(x\right)$-biharmonic
• generalized Sobolev and Lebesgue space
• Navier boundary conditions
• mountain pass theorem
• Ekeland variational principle

AMS Subject Classifications:

• 35D05
• 35J60
• 35J70
• 58E05

Acknowledgment

I would like to thank the anonymous referee for a series of careful reading, remarks, useful comments and valuable suggestions that enabled us essentially to improve this manuscript.

Disclosure statement

No potential conflict of interest was reported by the author(s).