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Global bifurcation for the Hénon problem

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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  • 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.

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


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