Choquard equations under confining external potentials

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• 作者：Jean Van Schaftingen ; Jiankang Xia
• 关键词：Nonlocal semilinear elliptic problem ; Weighted Sobolev embedding theorem ; Groundstate ; Fountain Theorem ; Least Action Nodal Solution
• 刊名：Nonlinear Differential Equations and Applications NoDEA
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：24
• 期：1
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Analysis;
• 出版者：Springer International Publishing
• ISSN：1420-9004
• 卷排序：24

文摘

We consider the nonlinear Choquard equation $$\begin{aligned} -{\Delta }u+V u=\bigl (I_\alpha *\vert u\vert ^p\bigr )\vert u\vert ^{p-2}u \quad \text { in } \; \mathbb {R}^N \end{aligned}$$where \(N\ge 1\), \(I_\alpha \) is the Riesz potential integral operator of order \(\alpha \in (0, N)\) and \(p > 1\). If the potential \( V \in C (\mathbb {R}^N; [0,+\infty )) \) satisfies the confining condition $$\begin{aligned} \liminf \limits _{\vert x\vert \rightarrow +\infty }\frac{V(x)}{1+\vert x\vert ^{\frac{N+\alpha }{p}-N}}=+\infty , \end{aligned}$$and \(\frac{1}{p} > \frac{N - 2}{N + \alpha }\), we show the existence of a groundstate, of an infinite sequence of solutions of unbounded energy and, when \(p \ge 2\) the existence of least energy nodal solution. The constructions are based on suitable weighted compact embedding theorems. 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Appl. 431(2), 935–954 (2015)MathSciNetCrossRefMATHGoogle ScholarCopyright information© Springer International Publishing 2017Authors and AffiliationsJean Van Schaftingen1Email authorView author's OrcID profileJiankang Xia2View author's OrcID profile1.Institut de Recherche en Mathématique et PhysiqueUniversité Catholique de LouvainLouvain-la-NeuveBelgium2.Chern Institute of Mathematics and LPMCNankai UniversityTianjinChina About this article CrossMark Publisher Name Springer International Publishing Print ISSN 1021-9722 Online ISSN 1420-9004 About this journal Reprints and Permissions Article actions .buybox { margin: 16px 0 0; position: relative; } .buybox { font-family: Source Sans Pro, Helvetica, Arial, sans-serif; font-size: 14px; font-size: .875rem; } .buybox { zoom: 1; } .buybox:after, .buybox:before { content: ''; display: table; } .buybox:after { clear: both; } /*---------------------------------*/ .buybox .buybox__header { border: 1px solid #b3b3b3; border-bottom: 0; padding: 8px 12px; 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