Compact Embeddings, Eigenvalue Problems, and subelliptic Brezis-Nirenberg equations involving singularity on stratified Lie groups

TL;DR

This paper investigates eigenvalue problems and existence of solutions for fractional p-sub-Laplacian equations on stratified Lie groups, introducing new variational and analytical techniques applicable even to the Heisenberg group.

Contribution

It introduces novel results on eigenvalues and solutions for fractional subelliptic equations on stratified Lie groups, including positivity, simplicity, and existence of multiple solutions.

Findings

First eigenfunction is positive and unique.

Existence of at least two weak solutions.

Boundedness of positive solutions via Moser iteration.

Abstract

The purpose of this paper is twofold: first we study an eigenvalue problem for the fractional $p$-sub-Laplacian over the fractional Folland-Stein-Sobolev spaces on stratified Lie groups. We apply variational methods to investigate the eigenvalue problems. We conclude the positivity of the first eigenfunction via the strong minimum principle for the fractional $p$-sub-Laplacian. Moreover, we deduce that the first eigenvalue is simple and isolated. Secondly, utilising established properties, we prove the existence of at least two weak solutions via the Nehari manifold technique to a class of subelliptic singular problems associated with the fractional $p$-sub-Laplacian on stratified Lie groups. We also investigate the boundedness of positive weak solutions to the considered problem via the Moser iteration technique. The results obtained here are also new even for the case $p=2$ with $\mathbb{G}$ being the Heisenberg group.