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-convexnonlinearities term $left{begin{array}{l};-triangle u;+;u;+;etavarphi u;=;lambda f(x);u^{q;-1}+;u^5;in;mathbb{R}^3,-trianglevarphi;=u^2;,;in;mathbb{R}^3,end{array}right.$ (0.1) where 1 0 is a real parameter and $f;in;L^frac6{6;-;q};(mathbb{R}^3)$ is a nonzero nonnegative function. Using thevariational method, we obtain that there exists a positive constant λ∗ > 0 such that for all λ ∈ (0, λ∗), the system hasat least two positive solutions.