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

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 $0$ ($\underset{\left|x\right|\to 0}{lim}u\left(x\right)∕E\left(x\right)\in \left(0,\infty \right)$, where $E$ denotes the fundamental solution of the Laplacian), or (3) $\underset{\left|x\right|\to 0}{lim}|x{|}^{\vartheta }u\left(x\right)=\lambda$, 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.