On the spectrum of Robin boundary p-Laplacian problem

Abstract

Abstract 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 λn↗∞. To this end we endow W1,p(Ω) with a norm invoking the trace and use the duality mapping on W1,p (Ω) to apply mini-max arguments on C1-manifold.