Singular limit and exact decay rate of a nonlinear elliptic equation

For any $n\ge 3$, $0<m\le (n-2)\mathrm{/}n$, and constants $\eta >0$, $\beta >0$, $\alpha$, satisfying $\alpha \le \beta (n-2)\mathrm{/}m$, 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 $0<m<(n-2)\mathrm{/}n$, $n\ge 3$, and $\alpha =2\beta \mathrm{/}(1-m)>0$, we prove that the radially symmetric solution $v$ of the above elliptic equation satisfies $\lim_{|x|\to\infty}\frac{|x|^2v(x)^{1-m}}{\log |x|} =\frac{2(n-1)(n-2-nm)}{\beta(1-m)}$\mathrm{.}Inparticularwhen$m=\frac{n-2}{n+2}$,$n\ge 3$,and$\alpha=2\beta/(1-m)>0$,themetric$g_{ij}=v^{\frac{4}{n+2}}dx^2$ is the steady soliton solution of the Yamabe flow on ${\mathbb{R}}^n$ and we obtain $\lim_{|x|\to\infty}\frac{|x|^2v(x)^{1-m}}{\log |x|}=\frac{(n-1)(n-2)}{\beta}$\mathrm{.}When$0<m<(n-2)/n$,$n\ge 3$,and$2\beta/(1-m)>\max (\alpha,0)$,weprovethat$\lim_{|x|\to\infty}|x|^{\alpha/\beta}v(x)=A$forsomeconstant$A>0$\mathrm{.}For$\beta>0$or$\alpha=0$,weprovethattheradiallysymmetricsolution$v^{(m)}$ of the above elliptic elliptic equation converges uniformly on every compact subset of ${\mathbb{R}}^n$ 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$m\to 0$.