Weighted Steklov Problem Under Nonresonance Conditions

Abstract

We deal with the existence of weak solutions of the nonlinear problem $−\Delta_{p}u + V |u|^{p−2}u = 0$ 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 conditions, existence results for the problem.