Boundary singularities of positive solutions of quasilinear Hamilton-Jacobi equations

2014/12/15 by Bidaut-Véron, Marie-Françoise, Garcia-Huidobro, Marta, Véron, Laurent
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1412.4613

Abstract

We study the boundary behaviour of the solutions of (E) -Δ_p u+|∇ u|^q=0 in a domain Ω⊂ ℝ^N, when N≥ 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 ∂Ω and that these solutions belong to two different classes of singular solutions. If q_*≤ 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.

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