Radial solutions of scaling invariant nonlinear elliptic equations with mixed reaction terms

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

doi:10.3934/dcds.2020067

Article Contents

2020, Volume 40, Issue 2: 933-982. Doi: 10.3934/dcds.2020067

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Radial solutions of scaling invariant nonlinear elliptic equations with mixed reaction terms

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 de Chile

Received: March 2019

Revised: August 2019

Published: November 2019

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Abstract

We study global properties of positive radial solutions of $ -\Delta u = u^p+M\left |{\nabla u}\right |^{\frac{2p}{p+1}} $ in $ \mathbb R^N $ where $ p>1 $ and $ M $ is a real number. We prove the existence or the non-existence of ground states and of solutions with singularity at $ 0 $ according to the values of $ M $ and $ p $.

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Mathematics Subject Classification: 35J62, 35B08, 68?4.

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Figure 1. $ M>0 $, $ K>0 $

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Figure 2. $ M<0 $, $ K\geq 0 $

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Figure 3. $ -\mu^*<M<0 $, $ K<0 $

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Figure 4. $ M = -\mu^* $, $ K<0 $

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Figure 5. $ M<-\mu^* $, $ K<0 $

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