# Singular periodic solutions to a critical equation in the Heisenberg group

## TL;DR

This paper constructs singular positive solutions to a critical nonlinear equation on the Heisenberg group, exhibiting periodicity and homogeneity, extending the understanding of such equations in sub-Riemannian geometry.

## Contribution

It introduces a method to construct singular periodic solutions on the Heisenberg group using Lyapunov-Schmidt reduction, based on periodization of known solutions.

## Key findings

- Existence of singular solutions with specific homogeneity
- Solutions exhibit periodicity under dilation
- Extension of Fowler solutions to the Heisenberg group

## Abstract

We construct positive solutions to the equation $$-\Delta_{\mathbf{H}^n} u = u^{\frac{Q+2}{Q-2}}$$ on the Heisenberg group, singular in the origin, similar to the Fowler solutions of the Yamabe equations on $\mathbf{R}^n$. These satisfy the homogeneity property $u\circ\delta_T=T^{-\frac{Q-2}{2}}u$ for some $T$ large enough, where $Q=2n+2$ and $\delta_T$ is the natural dilation in $\mathbf{H}^n$. We use the Lyapunov-Schmidt method applied to a family of approximate solutions built by periodization from the global regular solution classified by Jerison and Lee.