{"title":"Saddle solutions for the planar Schrödinger–Poisson system with exponential growth","authors":"Liying Shan, Wei Shuai","doi":"10.1007/s00030-024-00980-1","DOIUrl":null,"url":null,"abstract":"<p>In this paper, we are interested in the following planar Schrödinger–Poisson system </p><span>$$\\begin{aligned} \\left\\{ \\begin{array}{ll} -\\Delta u+a(x)u+2\\pi \\phi u=|u|^{p-2}ue^{\\alpha _0|u|^\\gamma }, \\ {} &{} x\\in {\\mathbb {R}}^2,\\\\ \\Delta \\phi =u^2,\\ {} &{} x\\in {\\mathbb {R}}^2, \\end{array} \\right. \\end{aligned}$$</span>(0.1)<p>where <span>\\(p>2\\)</span>, <span>\\(\\alpha _0>0\\)</span> and <span>\\(0<\\gamma \\le 2\\)</span>, the potential <span>\\(a:{\\mathbb {R}}^2\\rightarrow {\\mathbb {R}}\\)</span> is invariant under the action of a closed subgroup of the orthogonal transformation group <i>O</i>(2). As a consequence, we obtain infinitely many saddle type nodal solutions for equation (0.1) with their nodal domains meeting at the origin if <span>\\(0<\\gamma <2\\)</span> and <span>\\(p>2\\)</span>. Furthermore, in the critical case <span>\\(\\gamma =2\\)</span> and <span>\\(p>4\\)</span>, we prove that equation (0.1) possesses a positive solution which is invariant under the same group action.</p>","PeriodicalId":501665,"journal":{"name":"Nonlinear Differential Equations and Applications (NoDEA)","volume":"10 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-07-15","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/s00030-024-00980-1","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
In this paper, we are interested in the following planar Schrödinger–Poisson system
where \(p>2\), \(\alpha _0>0\) and \(0<\gamma \le 2\), the potential \(a:{\mathbb {R}}^2\rightarrow {\mathbb {R}}\) is invariant under the action of a closed subgroup of the orthogonal transformation group O(2). As a consequence, we obtain infinitely many saddle type nodal solutions for equation (0.1) with their nodal domains meeting at the origin if \(0<\gamma <2\) and \(p>2\). Furthermore, in the critical case \(\gamma =2\) and \(p>4\), we prove that equation (0.1) possesses a positive solution which is invariant under the same group action.