Classification of solutions for some elliptic system

In this paper, we classify the solution of the following elliptic system -Δu(x)=e3v(x),x∈R4,(-Δ)2v(x)=u(x)4,x∈R4.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle -\Delta u(x)=e^{3v(x)}, &{}\quad x\in {\mathbb {R}}^4, \\ \\ \displaystyle (-\Delta )^2v(x)=u(x)^4, &{}\quad x\in {\mathbb {R}}^4. \end{array} \right. \end{aligned}$$\end{document}Under some assumptions, we will show that the solution has the following form u(x)=C1(ε)ε2+|x-x0|2,v(x)=lnC2(ε)ε2+|x-x0|2,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} u(x)=\frac{C_1(\varepsilon )}{\varepsilon ^2+|x-x_0|^2},\ v(x)=\ln \frac{C_2(\varepsilon )}{\varepsilon ^2+|x-x_0|^2}, \end{aligned}$$\end{document}where C1,C2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_1,C_2$$\end{document} are two positive constants depending only on ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} and x0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_0$$\end{document} is a fixed point in R4.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^4.$$\end{document}