Compactness of Solutions to Sub-Elliptic Equations with Potential on the Heisenberg Group

TL;DR

This paper studies the compactness of solutions to a critical sub-elliptic equation with potential on the Heisenberg group, revealing conditions for solution set compactness and behavior at blow-up points.

Contribution

It establishes compactness criteria for solutions under non-degeneracy conditions and characterizes blow-up behavior in the Heisenberg group setting.

Findings

Solution set is compact under certain potential conditions.

Blow-up solutions require potential and sub-Laplacian to vanish at the blow-up point.

Overcomes geometric challenges of the Heisenberg group in analysis.

Abstract

In this paper, we investigate the compactness of nonnegative solutions to a critical sub-elliptic equation with a nonnegative potential on the Heisenberg group. We establish that the solution set is compact provided the potential satisfies certain non-degeneracy conditions. Moreover, we show that if a sequence of solutions blows up, both the potential and its sub-Laplacian must vanish at the blow-up point. Our analysis overcomes the inherent geometric and analytical challenges posed by the Heisenberg group, including the degeneracy of the sub-Laplacian, its non-commutative structure, and the anisotropic dilation symmetry.