Existence of semiclassical solutions for some critical Schrödinger–Poisson equations with potentials

In this paper we are interested in the existence of semiclassical solutions for the critical Schrödinger–Poisson equations with multiple potentials. Under suitable assumptions on the potentials, we are able to prove the existence of semiclassical solutions (uε,ϕε) with maximum points xε of uε concentrating at a special set CP characterized by some of the potentials. Moreover, for any sequence xε→x0∈CP, by setting vε(x)≔uε(εx+xε), the ground solutions vε convergence strongly in H1(R3) to a ground state solution v of −Δv+V(x0)v=P(x0)g(v)+Q(x0)|v|4v,inR3.