The Existence of Ground State Solutions for a Schrödinger–Bopp–Podolsky System with Convolution Nonlinearity

In this paper, we consider the following Schrödinger–Bopp–Podolsky system with convolution nonlinearity: -Δu+Vxu+ϕu=Iα∗Fufu,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}$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u+V\left( x \right) u+\phi u=\left( I_{\alpha }*F\left( u \right) \right) f\left( u \right) , \;\;&{} \text{ in } \ \ {\mathbb {R}}^3,\\ -\Delta \phi +a^{2}\Delta ^{2} \phi =4\pi u^{2}, \;\;&{} \text{ in } \ \ {\mathbb {R}}^3, \end{array}\right. } \end{aligned}$$\end{document}where α∈0,2,Iα:R3→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in \left( 0,2\right) , I_\alpha :{\mathbb {R}}^{3} \overset{}{\rightarrow } {\mathbb {R}}$$\end{document} is the Riesz potential, V∈C(R3,[0,∞)),V∞=limx→∞V(x)>0,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}$$V\in {\mathcal {C}}({\mathbb {R}}^{3}, [0,\infty )), V_{\infty }=\lim _{\left| x \right| \rightarrow \infty }V(x) >0, f\in {\mathcal {C}}({\mathbb {R}}, {\mathbb {R}})$$\end{document} and F(t)=∫0tfsds\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F(t)=\int _{0}^{t} f\left( s \right) \textrm{d}s$$\end{document} satisfying limt→∞Fttσ=∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lim _{\left| t \right| \rightarrow \infty } \frac{F\left( t \right) }{\left| t \right| ^{\sigma } }=\infty $$\end{document}, where σ=α+64\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma =\frac{\alpha +6}{4}$$\end{document}. Through careful analysis of the nonlinear terms, we prove that the existence of ground state solutions and positive minimal energy solutions for the above system.