Existence of multiple positive solutions for Schrödinger–Poisson systems with critical growth

Abstract

In this paper, we are concerned with the existence, multiplicity and concentration of positive ground state solutions for the semilinear Schrodinger–Poisson system $$\left\{\begin{array}{ll}-\varepsilon^{2} \Delta u+a(x)u+\lambda\phi(x)u=b(x)f(u)+|u|^{4}u,&x\in\mathbb{R}^{3}, \\ -\varepsilon^{2} \Delta\phi=u^{2},\ u\in H^{1}(\mathbb{R}^{3}),&x\in\mathbb{R}^{3},\end{array}\right.$$ where $${\varepsilon > 0}$$ is a small parameter, f is a continuous, superlinear and subcritical nonlinearity, and $${\lambda\neq0}$$ is a real parameter. Suppose that a(x) has at least one global minimum and b(x) has at least one global maximum. We prove that there are two families of positive solutions for sufficiently small $${\varepsilon > 0}$$ , of which one is concentrating on the set of minimal points of a and the other on the sets of maximal points of b. Moreover, we obtain some sufficient conditions for the nonexistence of positive ground state solutions.