Steklov problems for the p−Laplace operator involving Lq-norm

In this paper, we are concerned with the study of the spectrum for the nonlinear Steklov problem of the form { Δpu=| u |p-2uin Ω,| ∇u |p-2∂u∂v=λ‖ u ‖q,∂Ωp-q| u |q-2uon ∂Ω, \left\{ {\matrix{{{\Delta _p}u = {{\left| u \right|}^{p - 2}}u} \hfill & {{\rm{in}}\,\Omega ,} \hfill \cr {{{\left| {\nabla u} \right|}^{p - 2}}{{\partial u} \over {\partial v}} = \lambda \left\| u \right\|_{q,\partial \Omega }^{p - q}{{\left| u \right|}^{q - 2}}u} \hfill & {{\rm{on}}\,\partial \Omega ,} \hfill \cr } } \right. where Ω is a smooth bounded domain in ℝ^N(N ≥ 1), λ is a real number which plays the role of eigenvalue and the unknowns u ∈ W^1,p(Ω). Using the Ljusterneck-Shnirelmann theory on C¹ manifold and Sobolev trace embedding we prove the existence of an increasing sequence positive of eigenvalues (λ_k)_k≥1, for the above problem. We then establish that the first eigenvalue is simple and isolated.