Weighted Steklov problem under nonresonance conditions

Abstract

We deal with the existence of weak solutions of the nonlinear problem $-\Delta_{p}u+V|u|^{p-2}u$ in a bounded smooth domain $\Omega\subset \mathbb{R}^{N}$ which is subject to the boundary condition $|\nabla u|^{p-2}\frac{\partial u}{\partial \nu}=f(x,u)$. Here $V\in L^{\infty}(\Omega)$ possibly exhibit both signs which leads to an extension of particular cases in literature and $f$ is a Carathéodory function that satisfies some additional conditions. Finally we prove, under and between nonresonance condtions, existence results for the problem.

Keywords

• Nonresonance
• $p$-Laplacian operator
• Sobolev trace embedding
• Steklov problem
• First nonprincipal eigenvalue