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Nodal solutions for the fractional Yamabe problem on Heisenberg groups

Published online by Cambridge University Press:  26 January 2019

Alexandru Kristály
Affiliation:
Department of Economics, Babeş-Bolyai University, Cluj-Napoca400591, Romania and Institute of Applied Mathematics, Óbuda University, Budapest 1034, Hungary (alexandrukristaly@yahoo.com; kristaly.alexandru@nik.uni-obuda.hu)

Abstract

We prove that the fractional Yamabe equation ${\rm {\cal L}}_\gamma u = \vert u \vert ^{((4\gamma )/(Q-2\gamma ))}u$ on the Heisenberg group ℍn has [n + 1/2] sequences of nodal (sign-changing) weak solutions whose elements have mutually different nodal properties, where ${\rm {\cal L}}_\gamma $ denotes the CR fractional sub-Laplacian operator on ℍn, Q = 2n + 2 is the homogeneous dimension of ℍn, and $\gamma \in \bigcup\nolimits_{k = 1}^n [k,((kQ)/Q-1)))$ . Our argument is variational, based on a Ding-type conformal pulling-back transformation of the original problem into a problem on the CR sphere S2n + 1 combined with a suitable Hebey-Vaugon-type compactness result and group-theoretical constructions for special subgroups of the unitary group U(n + 1).

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2019

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Footnotes

Dedicated to Professor Patrizia Pucci on the occasion of her 65th birthday.

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