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

Abstract

In this paper, we study the following critical growth Schr¨odinger–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 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.