Sign-changing solutions for Schrödinger–Bopp–Podolsky system with general nonlinearity

In this paper, the following Schrödinger–Bopp–Podolsky system is studied -Δu+V(x)u+ϕu=f(u),inR3,-Δϕ+a2Δ2ϕ=4πu2,inR3,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u+V(x)u+\phi u=f(u), &{}\mathrm{\ in\ } \mathbb {R}^3,\\ -\Delta \phi +a^2\Delta ^2\phi =4\pi u^2, &{}\mathrm{\ in\ }\mathbb {R}^3, \end{array}\right. } \end{aligned}$$\end{document}where a⩾0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a\geqslant 0$$\end{document}, under general assumptions of V and f, by using an abstract critical theorem, which derived from the minimax method and the method of invariant sets of descending flow, the existence of sign-changing solution for this system is obtained, in particular, a sequence of high energy sign-changing solutions is obtained if f is odd. Moreover, we also study the asymptotic behavior of the solution as a→0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a\rightarrow 0$$\end{document}, specifically, the sign-changing solution we find tends to the sign-changing solution of the classical Schrödinger–Poisson system as a→0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a\rightarrow 0$$\end{document}.