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

Abstract: In this paper we are interested in the existence of semiclassical states for the Choquard type equation $$ -\vr^2\Delta u +V(x)u =\Big(\int_{\R^N} \frac{G(u(y))}{|x-y|^{\mu}dy\Big)g(u)} \quad \mbox{in $\R^N$},</sup> $$ where $0&lt;\mu&lt;N$, N≥3N\geq3, $\vr$ is a positive parameter and GG is the primitive of gg which is of critical growth due to the Hardy--Littlewood--Sobolev inequality. The potential function V(x)V(x) is assumed to be nonnegative with V(x)=0V(x)=0 in some region of R<sup>N\R<sup>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.