Existence and multiplicity results for some Schrodinger-Poisson system with critical growth

In this paper we study the existence and multiplicity of positive solutions for the Schriidinger-Poisson system with critical growth: {-epsilon(2)Delta u + V (x)u = f (u) + vertical bar u vertical bar(3)u phi, x is an element of R-3, -epsilon(2)Delta phi = vertical bar u vertical bar(5), x is an element of R-3, u is an element of H-1 (R-3), u(x) > 0, x is an element of R-3, where epsilon > 0 is a parameter, V : R-3 -> R is a continuous function and f : R -> R is a C-1 function. Under a global condition for V we prove that the above problem has a ground state solution and relate the number of positive solutions with the topology of the set where V attains its minimum, by using variational methods. (C) 2020 Published by Elsevier Inc.