# Yamabe-type equations on Carnot groups

## TL;DR

This paper investigates Yamabe-type elliptic equations on Carnot groups, establishing existence results for solutions using variational methods, with a focus on subelliptic critical equations on the Heisenberg group.

## Contribution

It proves the existence of solutions for a class of Yamabe-type equations on Carnot groups, including the Heisenberg group, using variational techniques.

## Key findings

- Existence of at least one nontrivial solution for subelliptic critical equations.
- Application of variational methods to elliptic equations on Carnot groups.
- Results extend Yamabe problem solutions to sub-Riemannian settings.

## Abstract

This article is concerned with a class of elliptic equations on Carnot groups depending of one real positive parameter and involving a critical nonlinearity. As a special case of our results we prove the existence of at least one nontrivial solution for a subelliptic critical equation defined on a smooth and bounded domain $D$ of the {Heisenberg group} $\mathbb{H}^n=\mathbb{C}^n\times \mathbb{R}$. Our approach is based on pure variational methods and locally sequentially weakly lower semicontinuous arguments.