#### Vol. 8, No. 8, 2015

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

### Joshua Ching and Florica Cîrstea

Vol. 8 (2015), No. 8, 1931–1962
##### Abstract

We completely classify the behaviour near $0$, as well as at $\infty$ when $\Omega ={ℝ}^{N}$, of all positive solutions of $\Delta u={u}^{q}|\nabla u{|}^{m}$ in $\Omega \setminus \left\{0\right\}$, where $\Omega$ is a domain in ${ℝ}^{N}$ ($N\ge 2$) and $0\in \Omega$. Here, $q\ge 0$ and $m\in \left(0,2\right)$ satisfy $m+q>1$. Our classification depends on the position of $q$ relative to the critical exponent ${q}_{\ast }:=\left(N-m\left(N-1\right)\right)∕\left(N-2\right)$ (with ${q}_{\ast }=\infty$ if $N=2$). We prove the following: if $q<{q}_{\ast }$, then any positive solution $u$ has either (1) a removable singularity at $0$, or (2) a weak singularity at $0$ ($\underset{|x|\to 0}{lim}u\left(x\right)∕E\left(x\right)\in \left(0,\infty \right)$, where $E$ denotes the fundamental solution of the Laplacian), or (3) $\underset{|x|\to 0}{lim}|x{|}^{\vartheta }u\left(x\right)=\lambda$, where $\vartheta$ and $\lambda$ are uniquely determined positive constants (a strong singularity). If $q\ge {q}_{\ast }$ (for $N>2$), then $0$ is a removable singularity for all positive solutions. Furthermore, for any positive solution in ${ℝ}^{N}\setminus \left\{0\right\}$, we show that it is either constant or has a nonremovable singularity at $0$ (weak or strong). The latter case is possible only for $q<{q}_{\ast }$, where we use a new iteration technique to prove that all positive solutions are radial, nonincreasing and converging to any nonnegative number at $\infty$. This is in sharp contrast to the case of $m=0$ and $q>1$, when all solutions decay to $0$. Our classification theorems are accompanied by corresponding existence results in which we emphasise the more difficult case of $m\in \left(0,1\right)$, where new phenomena arise.

##### Keywords
nonlinear elliptic equations, isolated singularities, Leray–Schauder fixed point theorem, Liouville-type result
##### Mathematical Subject Classification 2010
Primary: 35J25
Secondary: 35B40, 35J60