Semiclassical states for Choquard type equations with critical growth: critical frequency case * *Minbo Yang is the corresponding author who is partially supported by NSFC (11971436, 12011530199) and ZJNSF (LD19A010001); Yanheng Ding is partially supported by NSFC (11871242); Fashun Gao is partially supported by NSFC (11901155).

Abstract

In this paper we are interested in the existence of semiclassical states for the Choquard type equation where 0 < μ < N, N ⩾ 3, ɛ 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 , 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.