Nodal solutions for the fractional Yamabe problem on Heisenberg groups

TL;DR

This paper establishes the existence of multiple sign-changing solutions to the fractional Yamabe problem on Heisenberg groups using variational methods and symmetry group techniques.

Contribution

It introduces new variational methods and symmetry group constructions to find nodal solutions for the fractional Yamabe equation on Heisenberg groups.

Findings

Existence of multiple nodal solutions with different nodal properties.

Application of a Ding-type transformation to relate the problem to the CR sphere.

Development of a compactness result and group-theoretical tools for the analysis.

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 ${\bf U}(n+1).$