**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

### PDE Seminar

# 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 $\Omega \backslash \left\{0\right\}$, where
$\Omega $ denotes a
domain in ${\mathbb{R}}^{N}$
($N\ge 2$)
containing $0$.
We obtain a complete classification of the behaviour near
$0$ (as well
as at $\infty $ if
$\Omega ={\mathbb{R}}^{N}$) for all
positive ${C}^{1}\left(\Omega \backslash \left\{0\right\}\right)$
distribution solutions, together with corresponding existence results. When
$\Omega ={\mathbb{R}}^{N}$,
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
$\underset{\left|x\right|\to 0}{lim}u\left(x\right)\u2215E\left(x\right)\in \left(0,\infty \right)$ or
$\underset{\left|x\right|\to 0}{lim}|x{|}^{\vartheta}u\left(x\right)=\lambda $, where
$\vartheta $ and
$\lambda $
are precisely determined positive constants. When
$\Omega ={\mathbb{R}}^{N}$, 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.