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

Marie-Françoise Bidaut-Véron; Laurent Véron

doi:10.3934/dcds.2022107

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2023, Volume 43, Issue 3&4: 1026-1051. Doi: 10.3934/dcds.2022107

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

Marie-Françoise Bidaut-Véron and
Laurent Véron^,

Institut Denis Poisson, CNRS UMR 7013, Université de Tours, France

^*Corresponding author: Laurent Véron
^*Corresponding author: Laurent Véron

A Juan-Luis por su 75 cumpleaños. Cuarenta y seis años de amistad, respeto y admiración

Received: January 2022

Revised: June 2022

Early access: August 2022

Published: March 2023

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Abstract

We study positive functions satisfying (E)$ \;-\Delta u+m|\nabla u|^q-u^p = 0 $ in a domain $ {\Omega} $ or in $ {\mathbb 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; these conditions are expressed in terms of Bessel capacities on the boundary. We also study removability of boundary singular sets, and solutions with an isolated singularity on $ \partial\Omega $. The different results depend on two critical exponents for $ p = p_c: = \frac{N+1}{N-1} $ and for $ q = q_c: = \frac{N+1}{N} $, and on the sign of $ q-\frac{2p}{p+1} $

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Mathematics Subject Classification: Primary: 35J62, 35J66, 35J75; Secondary: 31C15.

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