Decay rate and radial symmetry of the exponential elliptic equation

TL;DR

This paper investigates the decay rate and radial symmetry of solutions to a specific exponential elliptic PDE in higher dimensions, establishing asymptotic behavior and symmetry under certain conditions.

Contribution

It proves the decay rate of solutions at infinity and establishes radial symmetry for solutions satisfying particular growth and integrability conditions.

Findings

Solution decays like -2 log|x| at infinity

Radial symmetry holds under mild conditions

Conditions on parameters α and β for symmetry

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.