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In this paper we are interested in the existence of semiclassical states forthe 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 theprimitive of $g$ which is of critical growth due to theHardy--Littlewood--Sobolev inequality. The potential function $V(x)$ is assumedto be nonnegative with $V(x)=0$ in some region of $\R^N$, which means it is ofthe critical frequency case. Firstly we study a Choquard equation with doublecritical exponents and prove the existence and multiplicity of semiclassicalsolutions by the Mountain-Pass Theorem and the genus theory. Secondly weconsider a class of critical Choquard equation without lower perturbation, byestablishing a global Compactness lemma for the nonlocal Choquard equation, weprove the multiplicity of high energy semiclassical states by theLusternik--Schnirelman theory.

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