On the spectrum of Robin boundary p-Laplacian problem

Abstract

Abstract We study the following nonlinear eigenvalue problem with nonlinear Robin boundary condition { - Δ p u = λ | u | p - 2 u       i n     Ω , | ∇ u | p - 2 ∇ u . v + | u | p - 2 u = 0       o n     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 C 1-manifold.