# American Institute of Mathematical Sciences

doi: 10.3934/cpaa.2021065

## Positive solutions for Choquard equation in exterior domains

 1 School of mathematics and statistics, Hubei Normal University, Huangshi, 435002, China 2 School of mathematics and statistics, Wuhan University, Wuhan, 430072, China

* Corresponding author

Received  December 2020 Revised  February 2021 Published  April 2021

Fund Project: P. Chen was supported by the Research Foundation of Education Bureau of Hubei Province, China(Grant No.Q20192505). X. Liu was supported by the NSFC (Grant No.11771342)

This work concerns with the following Choquard equation
 $\begin{equation*} \begin{cases} -\Delta u+ u = (\int_{\Omega}\frac{u^2(y)}{|x-y|^{N-2}}dy)u &{\rm{in }}\; \Omega ,\\ u\in H_0^1(\Omega), \end{cases} \end{equation*}$
where
 $\Omega\subseteq \mathbb{R}^{N}$
is an exterior domain with smooth boundary. We prove that the equation has at least one positive solution by variational and toplogical methods. Moreover, we establish a nonlocal version of global compactness result in unbounded domain.
Citation: Peng Chen, Xiaochun Liu. Positive solutions for Choquard equation in exterior domains. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2021065
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