Positive solutions of fractional Schrödinger-Poisson systems involving critical nonlinearities with potential

In this paper, we study the existence of multiple positive solutions for a class of fractional Schrödinger-Poisson systems involving sign-changing potential and critical nonlinearities on an unbounded domain. By means of the Nehari manifold and Ljusternik-Schnirelmann category, we show how the coefficient $g(x)$ of the critical nonlinearity affects the number of positive solutions, and present a novel relationship between the number of positive solutions and the category of the global maximum set of $g(x)$. The new feature which distinguishes this paper from other related works lies in the fact that we focus on the case of $1 < q < 2$, which has not been presented in the literature.