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

Abstract

Let $n\geq 3$, $0\le m<\frac{n-2}{n}$, $\rho_1>0$, $\beta>\beta_0^{(m)}=\frac{m\rho_1}{n-2-nm}$, $\alpha_m=\frac{2\beta+\rho_1}{1-m}$ and $\alpha=2\beta+\rho_1$. For any $\lambda>0$, we prove the uniqueness of radially symmetric solution $v^{(m)}$ of $\La(v^m/m)+\alpha_m v+\beta x\cdot\nabla v=0$, $v>0$, in $\R^n\setminus\{0\}$ 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 $m\to 0^+$, $v^{(m)}$ converges uniformly in $C^2(K)$ for any compact subset $K$ of $\R^n\setminus\{0\}$ to the solution $v$ of $\La\log v+\alpha v+\beta x\cdot\nabla v=0$, $v>0$, in $\R^n\bs\{0\}$, 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 $(\R^n\setminus\{0\})\times (0,T)$ which blows up near $\{0\}\times (0,T)$ at the rate $|x|^{-\frac{\alpha_m}{\beta}}$ satisfies some mild growth condition on $(\R^n\setminus\{0\})\times (0,T)$, then as $m\to 0^+$, $u^{(m)}$ converges uniformly in $C^{2+\theta,1+\frac{\theta}{2}}(K)$ for some constant $\theta\in (0,1)$ and any compact subset $K$ of $(\R^n\setminus\{0\})\times (0,T)$ to the solution of $u_t=\La\log u$, $u>0$, in $(\R^n\setminus\{0\})\times (0,T)$. As a consequence of the proof we obtain existence of a unique radially symmetric solution $v^{(0)}$ of $\La \log v+\alpha v+\beta x\cdot\nabla v=0$, $v>0$, in $\R^n\setminus\{0\}$, which satisfies $\lim_{|x|\to 0}|x|^{\frac{\alpha}{\beta}}v(x)=\lambda^{-\frac{\rho_1}{\beta}}$.