Singularities and asymptotics of solutions of the Chandrasekhar-Hailton-Jacobi equation

TL;DR

This work analyzes the local behavior near isolated singularities of positive solutions to the elliptic equation $-\Delta u + m|\\nabla u|^q - e^{u} = 0$ in punctured and exterior domains, highlighting how the regime of $q$ (relative to $2$ and $\\frac{N}{N-1}$) drives distinct singularity phenomena. It develops sharp a priori estimates for gradients and solutions, including Keller–Osserman-type bounds and a distributional decomposition that may involve a Dirac mass at the singularity, and it proves monotonicity and integrability properties that underpin the classification of singularities. For the subquadratic range $1<q<2$, the paper provides a detailed singularity classification using the Huang–Takác extension of Simon-type methods and a cylinder representation to capture angular behavior via isotropy estimates. The results yield precise near-singularity asymptotics, offer tools for studying exterior-domain behavior, and contribute robust techniques for nonlinear elliptic PDEs with gradient terms and exponential nonlinearities.

Abstract

We study the local properties of positive solutions of the equation $-Δu+ m\abs{\nabla u}^q-e^{u}=0$ in a punctured domain $Ω\setminus\{0\}$ of $R^N$, $N\geq 2$, where $m$ is a positive parameter and $q>1$. We study particularly the existence of solutions with an isolated singularity and the local behaviour of such singular solutions. .

Theorems & Definitions (21)

• Theorem 1.1
• Corollary 1.2
• Corollary 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Theorem 1.9
• Lemma 2.1