Ground state and multiple solutions for the fractional Schrodinger-Poisson system with critical Sobolev exponent

In this paper, we study the following doubly singularly perturbed fractional Schrodinger Poisson system with critical Sobolev exponent {epsilon(2 alpha)(-Delta)(alpha)u + V (x)u + phi u = vertical bar u vertical bar(2)(-2)(alpha*) u + f(u) in R-N, epsilon(theta)(-Delta)(s/2) phi = gamma(s)u(2) R-N, where alpha is an element of (1/2, 1), N is an element of (2 alpha 4 alpha), s is an element of (N - 2 alpha, N), theta is an element of (0, s), f is a subcritical nonlinearity, epsilon is a small parameter, the positive potential V satisfies a local condition. By combining penalization techniques with Ljusternik-Schnirelmann theory, the number of positive solutions is estimated below by the topology of the set where the potential V attains its minimum. (C) 2017 Elsevier Ltd. All rights reserved