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Existence of Solutions to the Logarithmic Choquard Equations in High Dimensions

  • Qianqiao GuoEmail author
  • Jing Wu
Article
  • 27 Downloads

Abstract

In this paper, we consider the following logarithmic Choquard equation
$$\begin{aligned} - \,\Delta u + a(x)u + \lambda (\log |\cdot |*|u|{^2})u = b|u|^{p-2}u, ~~\ \text{ in }~\ {\mathbb {R}}^n, \end{aligned}$$
where \(n\ge 3, \lambda >0, 2<p<\frac{2n}{n-2}, a \in L^\infty ({\mathbb {R}}^n).\) For \(n=2, p>2\), this equation has been studied extensively. In this paper, we prove the existence of a mountain-pass solution and a ground state solution for \(n\ge 3\) by using the variational method.

Keywords

Logarithmic Choquard equation Mountain-pass solution Ground state solution 

Mathematics Subject Classification

35A15 35J20 35Q40 

Notes

Acknowledgements

The first author is partially supported by the National Natural Science Foundation of China (Grant No. 11571268).

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Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2019

Authors and Affiliations

  1. 1.Department of Applied MathematicsNorthwestern Polytechnical UniversityXi’anChina

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