Decay rate and radial symmetry of the exponential elliptic equation

Let $n\ge 3$, $\alpha$, $\beta \in \mathbb{R}$, and let $v$ be a solution $\mathrm{\Delta }v+\alpha e^v+\beta x\cdot \mathrm{\nabla }{e}^v=0$ in ${\mathbb{R}}^n$, which satisfies the conditions $\lim_{R\to\infty}\frac{1}{\log R}\int_{1}^{R}\rho^{1-n} (\int_{B_{\rho}}e^v,dx)d\rho\in (0,\infty)$and$|x|^2e^{v(x)}\le A_1$in$\R^n$\mathrm{.}Weprovethat$\frac{v(x)}{\log |x|}\to -2$as$|x|\to\infty$and$\alpha>2\beta$. As a consequence under a mild condition on $v$ we prove that the solution is radially symmetric about the origin.