Competing Potentials Effect on Positive Solutions to a Schrödinger-Poisson System

In this paper, we study the multiplicity and concentration of positive solutions for a Schrödinger-Poisson system involving sign-changing potential and the nonlinearity K(x)∣u∣^p−2u (2 < p < 4) in ℝ³. Such a problem cannot be studied by variational methods in a standard way, even by restricting its corresponding energy functional on the Nehari manifold since its (PS) sequence may not be bounded. By some new analytic techniques and the Ljusternik-Schnirelmann category theory, we relate the concentration and the number of positive solutions to the category of the global minima set of a suitable ground energy function. Furthermore, we investigate the asymptotic behavior of the solutions. In particular, we do not use Pohozaev equality in this work.