Nodal solutions for the fractional Yamabe problem on Heisenberg groups

Abstract

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

14 pages; to appear in Proceedings of the Royal Society of Edinburgh. Section A. Mathematics

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