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In this paper, we are concerned with the study of the spectrum for the nonlinear Steklov problem of the form  { Δ p u = | u | p - 2 u in   Ω , | ∇ u | p - 2 ∂ u ∂ v = λ ‖ u ‖ q , ∂ Ω p - q | u | q - 2 u on   ∂ Ω , \left\{ {\matrix{{{\Delta _p}u = {{\left| u \right|}^{p - 2}}u} \hfill &amp; {{\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 &amp; {{\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  C 1  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.

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

Steklov problems for the p−Laplace operator involving L^q -norm