Nodal solutions for the fractional Yamabe problem on Heisenberg groups

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 & x210d;(n) 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 & x210d;(n), Q = 2n + 2 is the homogeneous dimension of & x210d;(n), 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).