Singular limits and properties of solutions of some degenerate elliptic and parabolic equations

Let $ngeq 3$, $0le m<frac{n-2}{n}$, $rho_1>0$, $beta>beta_0^{(m)}=frac{mrho_1}{n-2-nm}$, $alpha_m=frac{2beta+rho_1}{1-m}$ and $alpha=2beta+rho_1$. For any $lambda>0$, we prove the uniqueness of radially symmetric solution $v^{(m)}$ of $La(v^m/m)+alpha_m v+beta xcdot abla v=0$, $v>0$, in $R^nsetminus{0}$ which satisfies $lim_{|x|to 0}|x|^{frac{alpha_m}{beta}}v^{(m)}(x)=lambda^{-frac{rho_1}{(1-m)beta}}$ and obtain higher order estimates of $v^{(m)}$ near the blow-up point $x=0$. We prove that as $mto 0^+$, $v^{(m)}$ converges uniformly in $C^2(K)$ for any compact subset $K$ of $R^nsetminus{0}$ to the solution $v$ of $Lalog v+alpha v+beta xcdot abla v=0$, $v>0$, in $R^nbs{0}$, which satisfies $lim_{|x|to 0}|x|^{frac{alpha}{beta}}v(x)=lambda^{-frac{rho_1}{beta}}$. We also prove that if the solution $u^{(m)}$ of $u_t=Delta (u^m/m)$, $u>0$, in $(R^nsetminus{0})times (0,T)$ which blows up near ${0}times (0,T)$ at the rate $|x|^{-frac{alpha_m}{beta}}$ satisfies some mild growth condition on $(R^nsetminus{0})times (0,T)$, then as $mto 0^+$, $u^{(m)}$ converges uniformly in $C^{2+theta,1+frac{theta}{2}}(K)$ for some constant $thetain (0,1)$ and any compact subset $K$ of $(R^nsetminus{0})times (0,T)$ to the solution of $u_t=Lalog u$, $u>0$, in $(R^nsetminus{0})times (0,T)$. As a consequence of the proof we obtain existence of a unique radially symmetric solution $v^{(0)}$ of $La log v+alpha v+beta xcdot abla v=0$, $v>0$, in $R^nsetminus{0}$, which satisfies $lim_{|x|to 0}|x|^{frac{alpha}{beta}}v(x)=lambda^{-frac{rho_1}{beta}}$.