On the spectrum of Robin boundary p-Laplacian problem

We study the following nonlinear eigenvalue problem with nonlinear Robin boundary condition

{-Δpu=λ|u|p-2u in Ω,|∇u|p-2∇u.v+|u|p-2u=0 on Γ.\left\{ {\matrix{ { - {\Delta _p}u = \lambda {{\left| u \right|}^{p - 2}}u\,\,\,in\,\,\Omega ,} \hfill \cr {{{\left| {\nabla u} \right|}^{p - 2}}\nabla u.v + {{\left| u \right|}^{p - 2}}u = 0\,\,\,on\,\,\Gamma .} \hfill \cr } } \right.

We successfully investigate the existence at least of one nondecreasing sequence of positive eigenvalues λ_n↗^∞. To this end we endow W^1,p(Ω) with a norm invoking the trace and use the duality mapping on W^1,p (Ω) to apply mini-max arguments on C¹-manifold.