New multiple solutions for a Schrödinger–Poisson system involving concave-convex nonlinearities

Abstract

In this paper, we study the following critical growth Schrö dinger-Poisson system with concave-convex nonlinearities term { −∆u + u + ηφu = λf (x)u q−1 + u 5 , in R 3 , −∆φ = u 2 , in R 3 , (0.1) where 1 < q < 2 , η ∈ R , λ > 0 is a real parameter and f ∈ L 6 6−q (R 3) is a nonzero nonnegative function. Using the variational method, we obtain that there exists a positive constant λ * > 0 such that for all λ ∈ (0, λ *) , the system has at least two positive solutions.