In this paper we consider the nonlinear Choquard equationu) is of critical growth due to the Hardy-Littlewood-Sobolev inequality and G(x, u) =ˆu 0 g(x, s)ds. Firstly, by assuming that the potential V (x) might be sign-changing, we study the existence of Mountain-Pass solution via a concentration-compactness principle for the Choquard equation. Secondly, under the conditions introduced by Benci and Cerami [7], we also study the existence of high energy solution by using a global compactness lemma for the nonlocal Choquard equation. 2010 Mathematics Subject Classification. 35J20, 35J60, 35A15.

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