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

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–Litt...

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Published in:Proceedings of the Royal Society of Edinburgh. Section A. Mathematics 2020-04, Vol.150 (2), p.921-954
Main Authors: Gao, Fashun, da Silva, Edcarlos D., Yang, Minbo, Zhou, Jiazheng
Format: Article
Language:English
Subjects:
Mountains
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Summary: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.
ISSN:0308-2105
1473-7124
DOI:10.1017/prm.2018.131