Singular limit and exact decay rate of a nonlinear elliptic equation

TL;DR

This paper investigates the existence and asymptotic decay rates of radially symmetric solutions to a class of nonlinear elliptic equations, with applications to Yamabe flow steady solitons, extending understanding of their behavior at infinity.

Contribution

It establishes existence results without phase plane methods and derives precise decay rates for solutions, including special cases related to Yamabe flow steady solitons.

Findings

Existence of radially symmetric solutions under specified conditions.

Exact decay rates of solutions at infinity, including logarithmic behavior.

Convergence of solutions to a limit as the parameter m approaches zero.

Abstract

For any $n\ge 3$, $0<m\le (n-2)/n$, and constants $\eta>0$, $\beta>0$, $\alpha$, satisfying $\alpha\le\beta(n-2)/m$, we prove the existence of radially symmetric solution of $\frac{n-1}{m}\Delta v^m+\alpha v +\beta x\cdot\nabla v=0$, $v>0$, in $\R^n$, $v(0)=\eta$, without using the phase plane method. When $0<m<(n-2)/n$, $n\ge 3$, and $\alpha=2\beta/(1-m)>0$, we prove that the radially symmetric solution $v$ of the above elliptic equation satisfies $\lim_{|x|\to\infty}\frac{|x|^2v(x)^{1-m}}{\log |x|} =\frac{2(n-1)(n-2-nm)}{\beta(1-m)}$. In particular when $m=\frac{n-2}{n+2}$, $n\ge 3$, and $\alpha=2\beta/(1-m)>0$, the metric $g_{ij}=v^{\frac{4}{n+2}}dx^2$ is the steady soliton solution of the Yamabe flow on $\R^n$ and we obtain $\lim_{|x|\to\infty}\frac{|x|^2v(x)^{1-m}}{\log |x|}=\frac{(n-1)(n-2)}{\beta}$. When $0<m<(n-2)/n$, $n\ge 3$, and $2\beta/(1-m)>\max (\alpha,0)$, we prove that $\lim_{|x|\to\infty}|x|^{\alpha/\beta}v(x)=A$ for some constant $A>0$. For $\beta>0$ or $\alpha=0$, we prove that the radially symmetric solution $v^{(m)}$ of the above elliptic elliptic equation converges uniformly on every compact subset of $\R^n$ to the solution $u$ of the equation $(n-1)\Delta\log u+\alpha u+\beta x\cdot\nabla u=0$, $u>0$, in $\R^n$, $u(0)=\eta$, as $m\to 0$.