Discrete and Continuous Dynamical Systems - Series S (DCDS-S)

Hopf fibration and singularly perturbed elliptic equations

Pages: 823 - 838, Volume 7, Issue 4, August 2014      doi:10.3934/dcdss.2014.7.823

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Bernhard Ruf - Dip. di Matematica, Universita degli Studi, Via Saldini 50, 20133 Milano, Italy (email)
P. N. Srikanth - T.I.F.R. CAM, Bangalore, 560065, India (email)

Abstract: In this article we show how the Hopf fibration can be used to generate special solutions of singularly perturbed elliptic equations on annuli. Indeed, by the Hopf fibration the equation can be reduced to a lower dimensional problem, to which known results on single (or multiple point) concentration can be applied. Reversing the reduction process, one obtains solutions concentrating on circles and spheres, which are given as the fibres of the Hopf fibration.

Keywords:  Hopf fibration, qualitative properties of solutions, concentration on manifolds, superlinear elliptic equation, supercritical growth.
Mathematics Subject Classification:  Primary: 35J61, 35J20; Secondary: 35B25.

Received: November 2013;      Revised: December 2013;      Available Online: February 2014.