{"title":"q-Laplace equation involving the gradient on general bounded and exterior domains","authors":"A. Razani, C. Cowan","doi":"10.1007/s00030-023-00900-9","DOIUrl":null,"url":null,"abstract":"<p>The existence of positive singular solutions of </p><span>$$\\begin{aligned} \\left\\{ \\begin{array}{lcc} -\\Delta _q u=(1+g(x))|\\nabla u|^p &{}\\quad \\text {in}&{} B_1,\\\\ u=0&{}\\quad \\text {on}&{} \\partial B_1, \\end{array} \\right. \\end{aligned}$$</span>(1)<p>is proved, where <span>\\(B_1\\)</span> is the unit ball in <span>\\({\\mathbb {R}}^N\\)</span>, <span>\\(N \\ge 3\\)</span>, <span>\\(2<q<N\\)</span>, <span>\\(\\frac{N(q-1)}{N-1}<p<q\\)</span> and <span>\\(g\\ge 0\\)</span> is a Hölder continuous function with <span>\\(g(0) = 0\\)</span>. Also, the existence of positive singular solutions of </p><span>$$\\begin{aligned} \\left\\{ \\begin{array}{lcc} -\\Delta _q u=|\\nabla u|^p &{}\\quad \\text {in}&{} \\Omega ,\\\\ u=0&{}\\quad \\text {on}&{} \\partial \\Omega . \\end{array} \\right. \\end{aligned}$$</span>(2)<p>is proved, where <span>\\(\\Omega \\)</span> is a bounded smooth domain in <span>\\({\\mathbb {R}}^N\\)</span>, <span>\\(N \\ge 3\\)</span>, <span>\\(2< q<N\\)</span> and <span>\\(\\frac{N(q-1)}{N-1}<p<q\\)</span>. Finally, the existence of a bounded positive classical solution of (2) with the additional property that <span>\\(\\nabla u(x) \\cdot x > 0\\)</span> for large |<i>x</i>| is proved, in the case of <span>\\(\\Omega \\)</span> an exterior domain <span>\\({\\mathbb {R}}^N\\)</span>, <span>\\(N\\ge 3\\)</span> and <span>\\(p >\\frac{N(q-1)}{N-1}\\)</span>.</p>","PeriodicalId":501665,"journal":{"name":"Nonlinear Differential Equations and Applications (NoDEA)","volume":"887 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2023-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Nonlinear Differential Equations and Applications (NoDEA)","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s00030-023-00900-9","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
is proved, where \(B_1\) is the unit ball in \({\mathbb {R}}^N\), \(N \ge 3\), \(2<q<N\), \(\frac{N(q-1)}{N-1}<p<q\) and \(g\ge 0\) is a Hölder continuous function with \(g(0) = 0\). Also, the existence of positive singular solutions of
is proved, where \(\Omega \) is a bounded smooth domain in \({\mathbb {R}}^N\), \(N \ge 3\), \(2< q<N\) and \(\frac{N(q-1)}{N-1}<p<q\). Finally, the existence of a bounded positive classical solution of (2) with the additional property that \(\nabla u(x) \cdot x > 0\) for large |x| is proved, in the case of \(\Omega \) an exterior domain \({\mathbb {R}}^N\), \(N\ge 3\) and \(p >\frac{N(q-1)}{N-1}\).