Decay rate and radial symmetry of the exponential elliptic equation

Abstract

Let $n\ge 3$, $\alpha$, $\beta \in \mathbb{R}$, 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 $|x{|}^2{e}^{v(x)}\le A_1$ in ${\mathbb{R}}^n$. 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.

arXiv comment15 pages