Existence and multiplicity of solutions for some Styklov problem involving p(x)-Laplacian operator

Abstract

In this paper, we consider a class of $p\left(x\right)$-Laplacian problems of the form: $\begin{array}{l}(-\mathrm{\Delta }{)}_{p\left(x\right)}u+a(x)|u{|}^{p\left(x\right)-2}u=f(x,u)\text{in }\mathrm{\Omega },\\ |\mathrm{\nabla }u{|}^{p\left(x\right)-2}\frac{\mathrm{\partial }u}{\mathrm{\partial }v}+b\left(x\right)|u{|}^{q\left(x\right)-2}u=g(x,u)\text{on }\mathrm{\partial }\mathrm{\Omega },\end{array}$ where $\mathrm{\Omega }\subset {\mathbb{R}}^N$, $N\ge 2$ is a bounded domain with Lipschitz boundary $\mathrm{\partial }\mathrm{\Omega },\left(\mathrm{\partial }/\mathrm{\partial }v\right)$ is outer unit normal derivative. The functions a, b, p, q, g and f are assumed to satisfy suitable assymptions. The existence and the multiplicity of solutions is obtained by using variational methods, and mountain pass lemma combined with Ekeland variational principle.

COMMUNICATED BY:

• M. A. Ragusa

Keywords:

• $p\left(x\right)$-Laplacian
• variational methods
• variable exponents
• Ekeland principle
• generalized Sobolev space

2010 AMS Subject Classifications:

• 31B30
• 35J35

Acknowledgments

The authors would like to thank the anonymous referees for the valuable suggestions and comments which improved the quality of this paper.

Disclosure statement

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