Boundary singularities of positive solutions of quasilinear Hamilton-Jacobi equations

TL;DR

This paper investigates the boundary behavior of positive solutions to a class of quasilinear Hamilton-Jacobi equations, identifying a critical exponent that determines the existence and classification of boundary singularities.

Contribution

It introduces a critical exponent for boundary singularities in quasilinear equations and classifies solutions into two types based on their singularity behavior.

Findings

Existence of solutions with isolated boundary singularities for certain q values.

Non-existence of such solutions when q exceeds a critical threshold.

Complete classification of boundary singular solutions into two types.

Abstract

We study the boundary behaviour of the solutions of (E) $-\Delta_p u+|\nabla u|^q=0$ in a domain $\Omega \subset \mathbb{R}^N$, when $N\geq p > q >p-1$. We show the existence of a critical exponent $q_* < p$ such that if $p-1 < q < q_*$ there exist positive solutions of (E) with an isolated singularity on $\partial\Omega$ and that these solutions belong to two different classes of singular solutions. If $q_*\leq q < p$ no such solution exists and actually any boundary isolated singularity of a positive solution of (E) is removable. We prove that all the singular positive solutions are classified according the two types of singular solutions that we have constructed.