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

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 $\left\{Q_i\left(x\right)\right|1\le i\le m\}$ 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 $v_{\epsilon }$ in the semiclassical limit via the Nehari manifold and concentration–compactness principle. Then we show that $v_{\epsilon }$ 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.