Qualitative properties of stationary solutions of the NLS on the Hyperbolic space without and with external potentials

Alessandro Selvitella

doi:10.3934/cpaa.2019118

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2019, Volume 18, Issue 5: 2663-2677. Doi: 10.3934/cpaa.2019118

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Qualitative properties of stationary solutions of the NLS on the Hyperbolic space without and with external potentials

Alessandro Selvitella^,

Department of Mathematics and Statistics, University of Ottawa, 150 Louis Pasteur Private (ON) K1N 7N5 Canada

^* Corresponding author
^* Corresponding author

Received: August 2018

Revised: October 2018

Published online: September 2019

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Abstract

In this paper, we prove some qualitative properties of stationary solutions of the NLS on the Hyperbolic space. First, we prove a variational characterization of the ground state and give a complete characterization of the spectrum of the linearized operator around the ground state. Then we prove some rigidity theorems and necessary conditions for the existence of solutions in weighted spaces. Finally, we add a slowly varying potential to the homogeneous equation and prove the existence of non-trivial solutions concentrating on the critical points of a reduced functional. The results are the natural counterparts of the corresponding theorems on the Euclidean space. We produce also the natural virial identity on the Hyperbolic space for the complete evolution, which however requires the introduction of a weighted energy, which is not conserved and so does not lead directly to finite time blow-up as in the Euclidean case.

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Mathematics Subject Classification: Primary: 35Q55; Secondary: 35J61, 58J05.

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