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 -\vr^2\Delta u +V(x)u =\Big(\int_{\R^N} \frac{G(u(y))}{|x-y|^\mu}dy\Big)g(u) \quad \mbox{in $\R^N$}, where $0<\mu<N$, $N\geq3$, $\vr$ 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 solutions by the Mountain-Pass Theorem 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.