Global bifurcation for the Hénon problem

Anna Lisa Amadori

doi:10.3934/cpaa.2020212

Article Contents

2020, Volume 19, Issue 10: 4797-4816. Doi: 10.3934/cpaa.2020212

This issue Previous Article On some elliptic equation in the whole euclidean space $ \mathbb{R}^2 $ with nonlinearities having new exponential growth condition Next Article On the strauss index of semilinear tricomi equation

Global bifurcation for the Hénon problem

Anna Lisa Amadori

Dipartimento di Scienze Applicate, Università di Napoli "Parthenope", Centro Direzionale di Napoli, Isola C4, 80143 Napoli, Italy

Received: September 2019

Revised: May 2020

Published: July 2020

The author is member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM)..

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Abstract

We prove the existence of nonradial solutions for the Hénon equation in the ball with any given number of nodal zones, for arbitrary values of the exponent $ \alpha $. For sign-changing solutions, the case $ \alpha = 0 $ -Lane-Emden equation- is included. The obtained solutions form global continua which branch off from the curve of radial solutions $ p\mapsto u_p $, and the number of branching points increases with both the number of nodal zones and the exponent $ \alpha $. The proof technique relies on the index of fixed points in cones and provides information on the symmetry properties of the bifurcating solutions and the possible intersection and/or overlapping between different branches, thus allowing to separate them in some cases.

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Mathematics Subject Classification: 35J91, 35B05, 35B32.

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