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

Abstract

Let $n\ge 3$, $0\le m<\frac{n-2}{n}$, ${\rho }_1>0$, $\beta >{\beta }_0^{(m)}=\frac{m{\rho }_1}{n-2-nm}$, ${\alpha }_m=\frac{2\beta +{\rho }_1}{1-m}$ and $\alpha =2\beta +{\rho }_1$. For any $\lambda >0$, we prove the uniqueness of radially symmetric solution $v^{(m)}$ of $\La(v^m/m)+α_m v+βx\cdot\nabla v=0$, $v>0$, in ${\mathbb{R}}^n\setminus \{0\}$ 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 ${\mathbb{R}}^n\setminus \{0\}$ to the solution $v$ of $\La\log v+αv+βx\cdot\nabla v=0$, $v>0$, 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 $({\mathbb{R}}^n\setminus \{0\})\times (0,T)$ which blows up near $\{0\}\times (0,T)$ at the rate $|x{|}^{-\frac{{\alpha }_m}{\beta }}$ satisfies some mild growth condition on $({\mathbb{R}}^n\setminus \{0\})\times (0,T)$, 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 $({\mathbb{R}}^n\setminus \{0\})\times (0,T)$ to the solution of $u_t=\La\log u$, $u>0$, in $({\mathbb{R}}^n\setminus \{0\})\times (0,T)$. As a consequence of the proof we obtain existence of a unique radially symmetric solution $v^{(0)}$ of $\La \log v+αv+βx\cdot\nabla v=0$, $v>0$, in ${\mathbb{R}}^n\setminus \{0\}$, which satisfies ${\lim }_{|x|\to 0}|x{|}^{\frac{\alpha }{\beta }}v(x)={\lambda }^{-\frac{{\rho }_1}{\beta }}$.

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