{"title":"Existence and mass concentration of standing waves for inhomogeneous NLS equation with a bounded potential","authors":"Tian Tian, Jun Wang, Xiaoguang Li","doi":"10.1007/s00030-024-00969-w","DOIUrl":null,"url":null,"abstract":"<p>This paper is concerned with the following minimization problem </p><span>$$\\begin{aligned} e_p(M)=\\inf \\{E_p(u):u \\in H^1(\\mathbb {R}^N),\\Vert u\\Vert ^2_{L^2}=M^2 \\}, \\end{aligned}$$</span><p>where energy functional <span>\\(E_p(u)\\)</span> is defined by </p><span>$$\\begin{aligned} E_p(u)=\\Vert \\nabla u \\Vert _{L^2}^2 +\\int _{\\mathbb {R}^N} V(x)|u |^2dx -\\frac{2}{p+2} \\int _{\\mathbb {R}^N}|x |^{-h} | u|^{p+2}dx \\end{aligned}$$</span><p>and <i>V</i> is a bounded potential. For <span>\\(0<p< p^*:=\\frac{4-2\\,h}{N}(0<h<\\min \\{2,N\\})\\)</span>, it is shown that there exists a constant <span>\\(M_0\\ge 0\\)</span>, such that the minimization problem exists at least one minimizer if <span>\\(M> M_0\\)</span>. When <span>\\(p=p^*,\\)</span> the minimization problem exists at least one minimizer if <span>\\(M\\in (M_{*},\\Vert Q_{p^*}\\Vert _{L^2}),\\)</span> where constant <span>\\(M_{*}\\ge 0\\)</span> and <span>\\(Q_{p^*}\\)</span> is the unique positive radial solution of <span>\\(-\\Delta u+u -| x|^{-h}|u |^{p^*} u=0,\\)</span> and under some assumptions on <i>V</i>, there is no minimizer if <span>\\(M\\ge \\Vert Q_{p^*}\\Vert _{L^2}\\)</span>. Moreover, when <span>\\(0<p<p^*,\\)</span> for fixed <span>\\(M> \\Vert Q_{p^*}\\Vert _{L^2}\\)</span>, we analyze the concentration behavior of minimizers as <span>\\(p \\nearrow p^* \\)</span>.</p>","PeriodicalId":501665,"journal":{"name":"Nonlinear Differential Equations and Applications (NoDEA)","volume":"71 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-06-11","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/s00030-024-00969-w","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 following minimization problem
and V is a bounded potential. For \(0<p< p^*:=\frac{4-2\,h}{N}(0<h<\min \{2,N\})\), it is shown that there exists a constant \(M_0\ge 0\), such that the minimization problem exists at least one minimizer if \(M> M_0\). When \(p=p^*,\) the minimization problem exists at least one minimizer if \(M\in (M_{*},\Vert Q_{p^*}\Vert _{L^2}),\) where constant \(M_{*}\ge 0\) and \(Q_{p^*}\) is the unique positive radial solution of \(-\Delta u+u -| x|^{-h}|u |^{p^*} u=0,\) and under some assumptions on V, there is no minimizer if \(M\ge \Vert Q_{p^*}\Vert _{L^2}\). Moreover, when \(0<p<p^*,\) for fixed \(M> \Vert Q_{p^*}\Vert _{L^2}\), we analyze the concentration behavior of minimizers as \(p \nearrow p^* \).