Positive and Sign-changing Solutions for Critical Schrödinger–Poisson Systems with Sign-changing Potential

In this paper, we investigate the following critical Schrödinger–Poisson system -Δu+V(x)u+K(x)ϕu=f(u)+|u|4u,x∈R3,-Δϕ=K(x)u2,x∈R3,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u+V(x)u+K(x)\phi u = f(u)+|u|^4u,\ \ \ &{}\ x \in \mathbb {R}^{3},\\ -\Delta \phi =K(x)u^2, \ \ \ &{}\ x \in \mathbb {R}^{3},\\ \end{array}\right. } \end{aligned}$$\end{document}where V(x) is a (possible) sign-changing potential, K(x) is a nonnegative function and the nonlinearity f∈C(R,R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\in \mathcal {C}(\mathbb {R},\mathbb {R})$$\end{document}. By using variational methods with a more general global compactness lemma, we obtain a positive least energy solution and a least energy sign-changing solution with exactly two nodal domains, and we also prove that the energy of least energy sign-changing solution is strictly larger than twice that of least energy solutions. Moreover, this paper further analyzes the exponential decay of the positive least energy solution given by Liu, Liao and Tang (Nonlinearity30 (2017), 899–911), and can be regarded as the supplementary work of it.