Existence of positive solution for the Choquard equation in exterior domain

Abstract

In this paper, we study the nonlinear Choquard equation $\{\begin{array}{ll}-\mathrm{\Delta }u+\lambda u=(I_2\ast |u{|}^p)|u{|}^{p-2}u& \mathrm{in}\mathrm{\Omega },\\ u\in H_0^1(\mathrm{\Omega }),\end{array}$ where $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is an unbounded domain, $\mathrm{\partial }\mathrm{\Omega }\ne \mathrm{\varnothing }$ is bounded, $\lambda \in {\mathbb{R}}^{+}$, p = 2 if N = 3, 4, 5 or $2<p<\frac{7}{3}$ if N = 3. By global compactness analysis, we show that the problem has at least one positive high energy solution.

Keywords:

• Choquard equation
• positive solution
• exterior domain

AMS Subject Classifications:

• 35A15
• 35J20

Acknowledgements

The author would like to thank Professor Yinbin Deng very much for stimulating discussions and helpful suggestions on the present paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The research of X. L. Yang was supported by the graduate education innovation funding [2019CXZZ082] from Central China Normal University.