SMS scnews item created by Daniel Daners at Thu 23 Oct 2014 0950
Type: Seminar
Distribution: World
Expiry: 27 Oct 2014
Calendar1: 27 Oct 2014
CalLoc1: AGR Carslaw 829
Auth: daners@como.maths.usyd.edu.au

# Existence and classification of singular solutions to nonlinear elliptic equations with a gradient term

### Ching

Joshua Ching
University of Sydney
Mon 27 October 2014 2-3pm, Carslaw 829 (AGR)

## Abstract

For $m\in \left(0,2\right)$ and $q>max\left\{0,1-m\right\}$, we consider the nonlinear elliptic equation $\Delta u={u}^{q}|\nabla u{|}^{m}$ in $\Omega \\left\{0\right\}$, where $\Omega$ denotes a domain in ${ℝ}^{N}$ ($N\ge 2$) containing $0$. We obtain a complete classification of the behaviour near $0$ (as well as at $\infty$ if $\Omega ={ℝ}^{N}$) for all positive ${C}^{1}\left(\Omega \\left\{0\right\}\right)$ distribution solutions, together with corresponding existence results. When $\Omega ={ℝ}^{N}$, any positive solution with a removable singularity at $0$ must be constant. The origin is a removable singularity for all positive solutions if and only if ${E}^{q}|\nabla E{|}^{m}\notin {L}^{1}\left({B}_{r}\left(0\right)\right)$ for any small $r>0$ (possible for $N\ge 3$), where ${B}_{r}\left(0\right)$ denotes any ball centred at $0$ with small radius $r>0$ and $E$ denotes the fundamental solution of the Laplacian. If ${E}^{q}|\nabla E{|}^{m}\in {L}^{1}\left({B}_{r}\left(0\right)\right)$ for small $r>0$, then any given positive solution has either removable singularity at $0$, or $\underset{|x|\to 0}{lim}u\left(x\right)∕E\left(x\right)\in \left(0,\infty \right)$ or $\underset{|x|\to 0}{lim}|x{|}^{\vartheta }u\left(x\right)=\lambda$, where $\vartheta$ and $\lambda$ are precisely determined positive constants. When $\Omega ={ℝ}^{N}$, any positive solution is radially symmetric and non-increasing with (possibly any) non-negative limit at $\infty$.

Check also the PDE Seminar page. Enquiries to Daniel Hauer or Daniel Daners.

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