Trace and boundary singularities of positive solutions of a class of quasilinear equations

TL;DR

This paper investigates the boundary behavior and singularities of positive solutions to a class of quasilinear equations involving gradient and power nonlinearities, establishing existence, removability, and singularity criteria based on capacity and critical exponents.

Contribution

It provides new conditions for existence and removability of solutions with boundary measures and singularities, using capacity theory and critical exponent analysis.

Findings

Existence of solutions with boundary measure data under capacity conditions.

Characterization of removable boundary singularities.

Analysis of isolated boundary singularities based on critical exponents.

Abstract

We study properties of positive functions satisfying (E) --$\Delta$u+m|$\nabla$u| q -- u p = 0 is a domain $\Omega$ or in R N + when p > 1 and 1 < q < 2. We give sufficient conditions for the existence of a solution to (E) with a nonnegative measure $\mu$ as boundary data, and these conditions are expressed in terms of Bessel capacities on the boundary. We also study removable boundary singularities and solutions with an isolated singularity on $\partial$$\Omega$. The different results depends on two critical exponents for p = p c := N +1 N --1 and for q = q c := N +1 N and on the position of q with respect to 2p p+1. Contents