{"title":"On the spectrum of Robin boundary p-Laplacian problem","authors":"A. E. Khalil","doi":"10.2478/mjpaa-2019-0020","DOIUrl":null,"url":null,"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.","PeriodicalId":36270,"journal":{"name":"Moroccan Journal of Pure and Applied Analysis","volume":"5 1","pages":"279 - 293"},"PeriodicalIF":0.0000,"publicationDate":"2019-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"3","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Moroccan Journal of Pure and Applied Analysis","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.2478/mjpaa-2019-0020","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"Mathematics","Score":null,"Total":0}
引用次数: 3
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.