Weighted Steklov problem under nonresonance conditions

Abstract

We deal with the existence of weak solutions of the nonlinear problem −∆pu + V |u| p−2 u = 0 in a bounded smooth domain Ω ⊂ R N which is subject to the boundary condition |∇u| p−2 ∂u ∂ν = f (x, u). Here V ∈ L ∞ (Ω) 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.