Boundary singularities of positive solutions of quasilinear Hamilton-Jacobi equations

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 $N\ge p>q>p-1$. We show the existence of a critical exponent $q_{\ast }<p$ such that if $p-1<q<q_{\ast }$ 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 $q_{\ast }\le q<p$ 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.

arXiv commentCalculus of Variations and Partial Differential Equations DOI 10.1007/s00526-015-0911-5