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Nodal solutions for the fractional Yamabe problem on Heisenberg groups
- Part of
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 150 / Issue 2 / April 2020
- Published online by Cambridge University Press:
- 26 January 2019, pp. 771-788
- Print publication:
- April 2020
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- Article
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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).
