Semiclassical solutions for critical Schr\"odinger-Poisson systems involving multiple competing potentials
Abstract
In this paper, a class of Schr\"odinger-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 \Qi(x)|1 i 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 in the semiclassical limit via the Nehari manifold and concentration-compactness principle. Then we show that v 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.