Semiclassical states for Choquard type equations with critical growth: critical frequency case

In this paper we are interested in the existence of semiclassical states for the Choquard type equation -epsilon(2)Delta u+V(x)u = (integral(RN)G(u(y))/vertical bar x-y vertical bar(mu)dy) g(u) in R-N, where < mu < N, N >= 3, epsilon is a positive parameter and G is the primitive of g which is of critical growth due to the Hardy-Littlewood-Sobolev inequality. The potential function V(x) is assumed to be nonnegative with V(x) = 0 in some region of R-N, which means it is of the critical frequency case. Firstly, we study a Choquard equation with double critical exponents and prove the existence and multiplicity of semiclassical states by the mountain-pass lemma and the genus theory. Secondly, we consider a class of critical Choquard equation without lower perturbation, by establishing a global compactness lemma for the nonlocal Choquard equation, we prove the multiplicity of high energy semiclassical states by the Lusternik-Schnirelman theory.