We establish some existence results for the Brezis-Nirenberg type problem of the nonlinear Choquard equationwhere Ω is a bounded domain of R N with Lipschitz boundary, λ is a real parameter, N ≥ 3, 2 * µ = (2N − µ)/(N − 2) is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality. (2000): 35J25, 35J60, 35A15 Mathematics Subject Classifications

In this paper we study the semiclassical limit for the singularly perturbed Choquard equationwhere 0 < µ < 3, ε is a positive parameter, V, Q are two continuous real function on R 3 and G is the primitive of g which is of critical growth due to the Hardy-Littlewood-Sobolev inequality. Under suitable assumptions on g, we first establish the existence of ground states for the critical Choquard equation with constant coefficients. Next we establish existence and multiplicity of semi-classical solutions and characterize the concentration behavior by variational methods.2010 Mathematics Subject Classification. 35J20,35J60, 35B33.

We consider the following nonlinear Choquard equation with Dirichlet boundary conditionwhere Ω is a smooth bounded domain of R N , λ > 0, N ≥ 3, 0 < µ < N and 2 * µ is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality. Under suitable assumptions on different types of nonlinearities f (u), we are able to prove some existence and multiplicity results for the equation by variational methods. (2000): 35J25, 35J60, 35A15 Mathematics Subject Classifications

We study the existence of ground states for the nonlinear Choquard equation driven by fractional Laplacian:where the nonlinearity satisfies the general Berestycki-Lions-type assumptions.

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.