Decay rate and radial symmetry of the exponential elliptic equation

Abstract

Let $n\geq 3$, $\alpha$, $\beta\in\mathbb{R}$, and let $v$ be a solution $\Delta v+\alpha e^v+\beta x\cdot\nabla e^v=0$ 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 $\R^n$. We prove that $\frac{v(x)}{\log |x|}\to -2$ 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.