Singular solutions of some elliptic equations involving mixed absorption-reaction

Marie-Françoise Bidaut-Véron; Marta Garcia-Huidobro; Laurent Véron

doi:10.3934/dcds.2022036

Article Contents

2022, Volume 42, Issue 8: 3861-3930. Doi: 10.3934/dcds.2022036

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Singular solutions of some elliptic equations involving mixed absorption-reaction

1.
Laboratoire de Mathématiques et Physique Théorique, Université de Tours, 37200 Tours, France
2.
Departamento de Matematicas, Pontifica Universidad Catolica de Chile, Casilla 307, Correo 2, Santiago, Chile

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

This article has been prepared with the support of the FONDECYT grant 1210241 for the three authors

Received: August 2021

Early access: April 2022

Published: August 2022

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Abstract

We study properties of nonnegative functions satisfying (E)$ \;-{\Delta} u+u^p-M|\nabla u|^q = 0 $ in a domain of $ {\mathbb R}^N $ when $ p>1 $, $ M>0 $ and $ 1<q<p $. We concentrate our analysis on the solutions of (E) with an isolated singularity, or in an exterior domain, or in the whole space. The existence of such solutions and their behaviours depend strongly on the values of the exponents $ p $ and $ q $ and in particular according to the sign of $ q-\frac{2p}{p+1} $, and when $ q = \frac{2p}{p+1} $, also on the value of the parameter $ M $ which becomes a key element. The description of the different behaviours is made possible by a sharp analysis of the radial solutions of (E).

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Mathematics Subject Classification: Primary: 53C35, 35J60-35J62; Secondary: 35J75, 34A34, 34C37.

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Figure 1. $ M>0 $, $ K<0\Longleftrightarrow p<\frac{N}{N-2} $

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Figure 2. $ M>0 $, $ K = 0\Longleftrightarrow p = \frac{N}{N-2} $

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Figure 3. $ M>m^* $, $ K>0\Longleftrightarrow p>\frac{N}{N-2} $

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Figure 4. $ M = m^* $, $ K>0\Longleftrightarrow p>\frac{N}{N-2} $

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Figure 5. $ 0<M<m^* $, $ K>0\Longleftrightarrow p>\frac{N}{N-2} $

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