Semiclassical solutions for critical Schrödinger–Poisson systems involving multiple competing potentials

Abstract In this paper, a class of Schrödinger–Poisson system involving multiple competing potentials and critical Sobolev exponent is considered. Such a problem cannot be studied using the same argument for the nonlinear term with only a positive potential, because the weight potentials set contains nonpositive, sign changing, and nonnegative elements. By introducing the ground energy function and subtle analysis, we first prove the existence of ground state solution in the semiclassical limit via the Nehari manifold and concentration–compactness principle. Then we show that converges to the ground state solution of the associated limiting problem and concentrates at a concrete set characterized by the potentials. At the same time, some properties for the ground state solution are also studied. Moreover, a sufficient condition for the nonexistence of the ground state solution is obtained.