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

Milestones

Received: 6 August 2019

Accepted: 11 October 2019

Published: 29 April 2020