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.
Categories: math.AP