Classification of solutions for some elliptic system

Abstract

In this paper, we classify the solution of the following elliptic system

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle -\Delta u(x)=e^{3v(x)}, &{}\quad x\in {\mathbb {R}}^4, \\ \\ \displaystyle (-\Delta )^2v(x)=u(x)^4, &{}\quad x\in {\mathbb {R}}^4. \end{array} \right. \end{aligned}$$

Under some assumptions, we will show that the solution has the following form

$$\begin{aligned} u(x)=\frac{C_1(\varepsilon )}{\varepsilon ^2+|x-x_0|^2},\ v(x)=\ln \frac{C_2(\varepsilon )}{\varepsilon ^2+|x-x_0|^2}, \end{aligned}$$

where $C_1,C_2$ are two positive constants depending only on $\varepsilon$ and $x_0$ is a fixed point in ${\mathbb {R}}^4.$