# Nonlinear elliptic equations on Carnot groups

## TL;DR

This paper investigates a class of nonlinear elliptic equations on Carnot groups, establishing the existence of solutions using variational methods, with specific results for subelliptic equations on the Heisenberg group.

## Contribution

It extends the analysis of elliptic equations on Carnot groups to include parameter-dependent cases with subcritical nonlinearities, providing new existence results.

## Key findings

- Existence of at least one nontrivial solution for subelliptic equations on the Heisenberg group.
- Application of variational methods to nonlinear elliptic equations on Carnot groups.
- Results applicable to equations depending on a positive parameter with subcritical nonlinearities.

## Abstract

This article concerns a class of elliptic equations on Carnot groups depending on one real positive parameter and involving a subcritical nonlinearity (for the critical case we refer to G. Molica Bisci and D. Repov\v{s}, Yamabe-type equations on Carnot groups, Potential Anal. 46:2 (2017), 369-383; arXiv:1705.10100 [math.AP]). As a special case of our results we prove the existence of at least one nontrivial solution for a subelliptic equation defined on a smooth and bounded domain $D$ of the Heisenberg group $\mathbb{H}^n=\mathbb{C}^n\times \mathbb{R}$. The main approach is based on variational methods.