Semiclassical solutions for critical Schrodinger-Poisson systems involving multiple competing potentials

In this paper, a class of Schr{o}dinger-Poisson system involving multiple competing potentials and critical Sobolev exponent is considered. Such a problem cannot be studied with the same argument of the nonlinear term with only a positive potential, because the weight potentials set ${Q_i(x)|1le i le m}$ 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 $v_varepsilon$ in the semiclassical limit via the Nehari manifold and concentration-compactness principle. Then we show that $v_varepsilon$ 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.