Existence of solutions for a p(x)-biharmonic problem under Neumann boundary conditions

Abstract

In this paper, we consider for a given smooth bounded domain Ω of ${\mathbb{R}}^N,$ $(N\ge 3)$, the following $p\left(x\right)$-biharmonic type problem $\begin{array}{r}\begin{array}{ll}\mathrm{\Delta }\left(\xi \right(x\left)|\mathrm{\Delta }u{|}^{p\left(x\right)-2}\mathrm{\Delta }u\right)+a\left(x\right)|u{|}^{p\left(x\right)-2}u=\lambda \frac{\mathrm{\partial }F}{\mathrm{\partial }u}(x,u)& \text{in}\mathrm{\Omega }\\ \frac{\mathrm{\partial }u}{\mathrm{\partial }n}=0& \text{on}\mathrm{\partial }\mathrm{\Omega }\\ \frac{\mathrm{\partial }}{\mathrm{\partial }n}\left(\xi \right(x\left)|\mathrm{\Delta }u{|}^{p\left(x\right)-2}\mathrm{\Delta }u\right)=0& \text{on}\mathrm{\partial }\mathrm{\Omega },\end{array}\end{array}$ where λ is a positive parameter, $p\in C_{+}\left(\overline{\mathrm{\Omega }}\right)$ with $p^{-}:=\underset{x\in \overline{\mathrm{\Omega }}}{inf}p\left(x\right)>1,$ ξ is a function which satisfies the condition $0<{\xi }_1\le \xi \left(x\right)\le {\xi }_2,$ $a\in L^{\mathrm{\infty }}\left(\mathrm{\Omega }\right)$ with $\underset{x\in \overline{\mathrm{\Omega }}}{essinf}a\left(x\right)>0$ and $F:\overline{\mathrm{\Omega }}\times \mathbb{R}⟶\mathbb{R}$ is a $C^1$ function. We prove the existence of a continuous family of eigenvalues in a neighbourhood of the origin, under some suitable conditions by using Ekeland's principle and variational method.

COMMUNICATED BY:

• S. Leonardi

KEYWORDS:

• $p\left(x\right)$-biharmonic operator
• Ekeland's variational principle
• generalized Sobolev spaces

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 35J65
• 35D05
• 35J60
• 35D30
• 35J58

Disclosure statement

No potential conflict of interest was reported by the authors.