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Existence of solutions for critical Choquard equations via the concentration-compactness method

Part of: Partial differential equations Elliptic equations and systems

Published online by Cambridge University Press:  26 January 2019

Fashun Gao ,
Edcarlos D. da Silva ,
Minbo Yang  and
Jiazheng Zhou
Fashun Gao
Affiliation:
Department of Mathematics and Physics, Henan University of Urban Construction, Pingdingshan467044, People's Republic of China (fsgao@zjnu.edu.cn)
Edcarlos D. da Silva
Affiliation:
IME C Universidade Federal de Goiás, 74001-970 Goiania, GO, Brazil (eddomingos@hotmail.com)
Minbo Yang
Affiliation:
Department of Mathematics, Zhejiang Normal University, Jinhua321004, People's Republic of China (mbyang@zjnu.edu.cn)
Jiazheng Zhou
Affiliation:
Departamento de Matemática, Universidade de Bras´ilia, 70910-900 Bras´ilia DF, Brazil (jiazzheng@gmail.com)
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Abstract

In this paper, we consider the nonlinear Choquard equation

$$-\Delta u + V(x)u = \left( {\int_{{\open R}^N} {\displaystyle{{G(u)} \over { \vert x-y \vert ^\mu }}} \,{\rm d}y} \right)g(u)\quad {\rm in}\;{\open R}^N, $$
where 0 < μ < N, N ⩾ 3, g(u) is of critical growth due to the Hardy–Littlewood–Sobolev inequality and $G(u)=\int ^u_0g(s)\,{\rm d}s$. Firstly, by assuming that the potential V(x) might be sign-changing, we study the existence of Mountain-Pass solution via a nonlocal version of the second concentration- compactness principle. Secondly, under the conditions introduced by Benci and Cerami , we also study the existence of high energy solution by using a nonlocal version of global compactness lemma.

Keywords

Critical Choquard equationHardy–Littlewood–Sobolev inequalityConcentration-Compactness principle

MSC classification

Primary: 35J20: Variational methods for second-order elliptic equations
Secondary: 35J60: Nonlinear elliptic equations 35A15: Variational methods
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 2 , April 2020 , pp. 921 - 954
Copyright
Copyright © Royal Society of Edinburgh 2019

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