Stability of positive solutions to biharmonic equations on Heisenberg group

TL;DR

This paper proves the existence and stability of positive solutions to a biharmonic equation on the Heisenberg group, expanding understanding of such equations in sub-Riemannian geometry.

Contribution

It establishes the first known results on positive solution existence and stability for biharmonic equations on the Heisenberg group.

Findings

Existence of positive solutions is proven.

Solutions are shown to be stable.

Results extend biharmonic equation theory to sub-Riemannian settings.

Abstract

In this note, we establish the existence of a positive solution and its stability to the following problem $$\Delta_{\mathbb{H}^n}^2u=a(\xi)u-f(\xi,u)\text{ in }\Omega, \,\,\, u|_{\partial\Omega} = 0 =\left.\Delta_{\mathbb{H}^n} u|_{\partial\Omega},$$ on Heisenberg group.