- LandOfFree
- Scientists
- Mathematics
- Analysis of PDEs
Details
Singular limit and exact decay rate of a nonlinear elliptic equation
Singular limit and exact decay rate of a nonlinear elliptic equation
2011-07-14
-
arxiv.org/abs/1107.2735v1
Mathematics
Analysis of PDEs
19 pages
Scientific paper
For any $n\ge 3$, $00$, $\beta>0$, $\alpha$, satisfying $\alpha\le\beta(n-2)/m$, 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 $00$, we prove that the radially symmetric solution $v$ of the above elliptic equation satisfies $\lim_{|x|\to\infty}\frac{|x|^2v(x)^{1-m}}{\log |x|} =\frac{2(n-1)(n-2-nm)}{\beta(1-m)}$. In particular when $m=\frac{n-2}{n+2}$, $n\ge 3$, and $\alpha=2\beta/(1-m)>0$, 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 $\lim_{|x|\to\infty}\frac{|x|^2v(x)^{1-m}}{\log |x|}=\frac{(n-1)(n-2)}{\beta}$. When $0\max (\alpha,0)$, we prove that $\lim_{|x|\to\infty}|x|^{\alpha/\beta}v(x)=A$ for some constant $A>0$. For $\beta>0$ or $\alpha=0$, 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 $(n-1)\Delta\log u+\alpha u+\beta x\cdot\nabla u=0$, $u>0$, in $\R^n$, $u(0)=\eta$, as $m\to 0$.
Affiliated with
Also associated with
No associations
LandOfFree
Say what you really think
Search LandOfFree.com for scientists and scientific papers. Rate them and share your experience with other people.
Rating
Singular limit and exact decay rate of a nonlinear elliptic equation does not yet have a rating.
At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Singular limit and exact decay rate of a nonlinear elliptic equation, we encourage you to share that experience with our LandOfFree.com community.
Your opinion is very important and Singular limit and exact decay rate of a nonlinear elliptic equation will most certainly appreciate the feedback.
Rate now
Profile ID: LFWR-SCP-O-224816
All data on this website is collected from public sources.
Our data reflects the most accurate information available at the time of publication.