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

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 ${ℝ}^N$ ($N\ge 2$) containing $0$. We obtain a complete classification of the behaviour near $0$ (as well as at $\infty$ if $\Omega ={ℝ}^N$) for all positive $C^1\left(\Omega \backslash \left\{0\right\}\right)$ 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 $\underset{\left|x\right|\to 0}{lim}u\left(x\right)∕E\left(x\right)\in \left(0,\infty \right)$ 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 $\Omega ={ℝ}^N$, any positive solution is radially symmetric and non-increasing with (possibly any) non-negative limit at $\infty$.