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

Abstract

AbstractIn 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 potentialV(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.