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Existence of solutions for critical Choquard equations via the concentration-compactness method
- Part of
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 150 / Issue 2 / April 2020
- Published online by Cambridge University Press:
- 26 January 2019, pp. 921-954
- Print publication:
- April 2020
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- Article
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In this paper, we consider the nonlinear Choquard equation
where 0 < μ < N, N ⩾ 3, g(u) is of critical growth due to the Hardy–Littlewood–Sobolev inequality and
$$-\Delta u + V(x)u = \left( {\int_{{\open R}^N} {\displaystyle{{G(u)} \over { \vert x-y \vert ^\mu }}} \,{\rm d}y} \right)g(u)\quad {\rm in}\;{\open R}^N, $$
$G(u)=\int ^u_0g(s)\,{\rm d}s$. 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.
