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Existence of Positive Ground-State Solution for Choquard-Type Equations

  • Tao Wang
Article
  • 167 Downloads

Abstract

In this paper, we are concerned with the following nonlocal problem
$$\begin{aligned} -\Delta u+u=q(x)\left( \int _{\mathbb {R}^N}\frac{q(y)|u(y)|^p}{|x-y|^{N-\alpha }}\mathrm{d}y\right) |u|^{p-2}u,\quad x\in \mathbb {R}^N, \end{aligned}$$
where \(N\ge 3, \alpha \in ((N-4)_+,N), 2\le p<\frac{N+\alpha }{N-2}\) and q(x) is a given potential. Using comparison arguments and variational approach, we obtain the existence of positive ground-state solution for the Choquard-type equations with some restrictions on the potential q.

Keywords

Choquard equation ground-state solution comparison principles variational approach 

Mathematics Subject Classification

35B09 35B51 35J20 

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Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.College of Mathematics and Computing ScienceHunan University of Science and TechnologyXiangtanPeople’s Republic of China
  2. 2.School of Mathematics and StatisticsCentral South UniversityChangshaPeople’s Republic of China

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