#### Vol. 305, No. 2, 2020

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Singular periodic solutions to a critical equation in the Heisenberg group

### Claudio Afeltra

Vol. 305 (2020), No. 2, 385–406
DOI: 10.2140/pjm.2020.305.385
##### Abstract

We construct positive solutions to the equation

 $-{\Delta }_{{ℍ}^{n}}u={u}^{\frac{Q+2}{Q-2}}$

on the Heisenberg group, singular in the origin, similar to the Fowler solutions of the Yamabe equations on ${ℝ}^{n}$. These satisfy the homogeneity property $u\circ {\delta }_{T}={T}^{-\left(Q-2\right)∕2}u$ for some $T$ large enough, where $Q=2n+2$ and ${\delta }_{T}$ is the natural dilation in ${ℍ}^{n}$. 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 (1988).

##### Keywords
subelliptic equations, perturbation methods
##### Mathematical Subject Classification 2010
Primary: 35H20, 35J20, 35J61, 35R03