{"title":"Existence of Normalized Solutions for Mass Super-Critical Quasilinear Schrödinger Equation with Potentials","authors":"Fengshuang Gao, Yuxia Guo","doi":"10.1007/s12220-024-01779-3","DOIUrl":null,"url":null,"abstract":"<p>This paper is concerned with the existence of normalized solutions to a mass-supercritical quasilinear Schrödinger equation: </p><span>$$\\begin{aligned} \\left\\{ \\begin{array}{ll} -\\Delta u-u\\Delta u^2+V(x)u+\\lambda u=g(u),\\hbox { in }{\\mathbb {R}}^N, \\\\ u\\ge 0, \\end{array}\\right. \\end{aligned}$$</span>(0.1)<p>satisfying the constraint <span>\\(\\int _{{\\mathbb {R}}^N}u^2=a\\)</span>. We will investigate how the potential and the nonlinearity effect the existence of the normalized solution. As a consequence, under a smallness assumption on <i>V</i>(<i>x</i>) and a relatively strict growth condition on <i>g</i>, we obtain a normalized solution for <span>\\(N=2\\)</span>, 3. Moreover, when <i>V</i>(<i>x</i>) is not too small in some sense, we show the existence of a normalized solution for <span>\\(N\\ge 2\\)</span> and <span>\\(g(u)={u}^{q-2}u\\)</span> with <span>\\(4+\\frac{4}{N}<q<2\\cdot 2^*\\)</span>.\n</p>","PeriodicalId":501200,"journal":{"name":"The Journal of Geometric Analysis","volume":"21 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"The Journal of Geometric Analysis","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s12220-024-01779-3","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
This paper is concerned with the existence of normalized solutions to a mass-supercritical quasilinear Schrödinger equation:
satisfying the constraint \(\int _{{\mathbb {R}}^N}u^2=a\). We will investigate how the potential and the nonlinearity effect the existence of the normalized solution. As a consequence, under a smallness assumption on V(x) and a relatively strict growth condition on g, we obtain a normalized solution for \(N=2\), 3. Moreover, when V(x) is not too small in some sense, we show the existence of a normalized solution for \(N\ge 2\) and \(g(u)={u}^{q-2}u\) with \(4+\frac{4}{N}<q<2\cdot 2^*\).