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

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.

arXiv comment40 pages. To appear in "Mathematische Annalen"

References in corpus (10)

• Local behavior of fractional $p$-minimizers
• The second eigenvalue of the fractional $p-$Laplacian
• Harnack inequality for fractional sub-Laplacians in Carnot groups
• Yamabe-type equations on Carnot groups
• Existence theorems of the fractional Yamabe problem
• Global existence and blow-up of solutions to porous medium equation and pseudo-parabolic equation, I. Stratified Groups
• Nonlinear elliptic equations on Carnot groups
• Nodal solutions for the fractional Yamabe problem on Heisenberg groups
• An integral type Brezis-Nirenberg problem on the Heisenberg group
• A comparison principle for nonlinear heat Rockland operators on graded groups

Cited by in corpus (5)

• On semilinear equations with free boundary conditions on stratified Lie groups
• Subelliptic Nonlocal Brezis-Nirenberg Problems on Stratified Lie Groups
• A nonlocal elliptic problem on a Heisenberg group
• On subelliptic equations on stratified Lie groups driven by singular nonlinearity and weak $L^1$ data
• A multiphase eigenvalue problem on a stratified Lie group