Singular limit and exact decay rate of a nonlinear elliptic equation

Abstract

For any $n\ge 3$, $0 0$, $β>0$, $α$, satisfying $α\leβ(n-2)/m$, we prove the existence of radially symmetric solution of $\frac{n-1}{m}Δv^m+αv +βx\cdot\nabla v=0$, $v>0$, in $\R^n$, $v(0)=η$, without using the phase plane method. When $0 0$, we prove that the radially symmetric solution $v$ of the above elliptic equation satisfies $\lim_{|x|\to\infty}\frac{|x|^2v(x)^{1-m}}{\log |x|} =\frac{2(n-1)(n-2-nm)}{β(1-m)}$. In particular when $m=\frac{n-2}{n+2}$, $n\ge 3$, and $α=2β/(1-m)>0$, 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 $\lim_{|x|\to\infty}\frac{|x|^2v(x)^{1-m}}{\log |x|}=\frac{(n-1)(n-2)}β$. When $0 \max (α,0)$, we prove that $\lim_{|x|\to\infty}|x|^{α/β}v(x)=A$ for some constant $A>0$. For $β>0$ or $α=0$, 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 $(n-1)Δ\log u+αu+βx\cdot\nabla u=0$, $u>0$, in $\R^n$, $u(0)=η$, as $m\to 0$.