Nonresonance between the first two eigenvalues for a Steklov problem

Abstract

In this paper, we study the solvability of the Steklov problem Δ p u = | u | p − 2 u in Ω , | ∇ u | p − 2 ∂ u ∂ ν = f ( x , u ) on ∂ Ω , under assumptions on the asymptotic behaviour of the quotients f ( x , s ) / | s | p − 2 s and p F ( x , s ) / | s | p which extends the classical results with Dirichlet boundary conditions that for a.e. x ∈ ∂ Ω , the limits at the infinity of these quotients lie between the first two eigenvalues.