A coupled Schrödinger system with critical exponent

For the coupled Schrödinger system -Δu+V1(x)u=μ1u3+βuv2inR4,-Δv+V2(x)v=βu2v+μ2v3inR4,u≥0,v≥0inR4,u,v∈D1,2(R4),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{aligned}&-\Delta u+V_1(x)u=\mu _1u^3+\beta uv^2\ \ \text{ in }\ {{\mathbb {R}}}^4,\\&-\Delta v+V_2(x)v=\beta u^2v+\mu _2v^3\ \ \,\text{ in }\ {{\mathbb {R}}}^4,\\&\ u\ge 0,\ v\ge 0\ \ \text{ in }\ {{\mathbb {R}}}^4,\ \ u, v\in D^{1,2}({{\mathbb {R}}}^4), \end{aligned}\right. \end{aligned}$$\end{document}where V1,V2∈L2(R4)∩Lloc∞(R4)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V_1, V_2\in L^2({{\mathbb {R}}}^4)\,\cap \, L_\mathrm{loc}^\infty ({{\mathbb {R}}}^4)$$\end{document} are nonnegative functions and μ1,μ2,β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _1, \mu _2, \beta $$\end{document} are positive constants, we prove that if β>max{μ1,μ2}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta >\max \{\mu _1, \mu _2\}$$\end{document}, |V1|L2(R4)+|V2|L2(R4)>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|V_1|_{L^2({{\mathbb {R}}}^4)}+|V_2|_{L^2({{\mathbb {R}}}^4)}>0$$\end{document}, and |V1|L2(R4)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|V_1|_{L^2({{\mathbb {R}}}^4)}$$\end{document} and |V2|L2(R4)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|V_2|_{L^2({{\mathbb {R}}}^4)}$$\end{document} are suitably small, a positive solution exists. This generalizes the well known result in [8] by Benci and Cerami on a scalar equation to the above system.