Nonresonance between the first two eigenvalues for a Steklov problem

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