Calculus of Variations and Geometric Measure Theory

C. Afeltra

Singular periodic solutions to a critical equation in the Heisenberg group

created by afeltra on 07 Aug 2019
modified on 11 Dec 2019



Inserted: 7 aug 2019
Last Updated: 11 dec 2019

Year: 2019

ArXiv: 1908.02264 PDF


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.