Positive solutions for Choquard equation in exterior domains

Peng Chen; Xiaochun Liu

doi:10.3934/cpaa.2021065

Article Contents

2021, Volume 20, Issue 6: 2237-2256. Doi: 10.3934/cpaa.2021065

This issue Previous Article Boundedness and asymptotic stability in a quasilinear two-species chemotaxis system with nonlinear signal production Next Article Fractional oscillon equations; solvability and connection with classical oscillon equations

Positive solutions for Choquard equation in exterior domains

Peng Chen^1, , and
Xiaochun Liu^2,

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
^* Corresponding author

Received: November 30, 2020

Revised: January 31, 2021

Early access: April 2021

Published: June 2021

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)

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Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 35J20, 35J60; Secondary: 35A16.

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