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Compact embeddings, eigenvalue problems, and subelliptic Brezis–Nirenberg equations involving singularity on stratified Lie groups

Sekhar Ghosh, Vishvesh Kumar (UGent) and Michael Ruzhansky (UGent)
(2024) MATHEMATISCHE ANNALEN. 388(4). p.4201-4249
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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 G being the Heisenberg group.
Keywords
‘Ghent Analysis & PDE center, 35R03, 35H20, 35P30, 22E30, 35R11, 35J75, POSITIVE SOLUTIONS, DIFFERENTIAL-OPERATORS, ELLIPTIC-EQUATIONS, HARNACK INEQUALITY, DIRICHLET PROBLEM, CRITICAL GROWTH, SUB-LAPLACIANS, P-LAPLACIAN, EXISTENCE, MULTIPLICITY

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MLA
Ghosh, Sekhar, et al. “Compact Embeddings, Eigenvalue Problems, and Subelliptic Brezis–Nirenberg Equations Involving Singularity on Stratified Lie Groups.” MATHEMATISCHE ANNALEN, vol. 388, no. 4, 2024, pp. 4201–49, doi:10.1007/s00208-023-02609-7.
APA
Ghosh, S., Kumar, V., & Ruzhansky, M. (2024). Compact embeddings, eigenvalue problems, and subelliptic Brezis–Nirenberg equations involving singularity on stratified Lie groups. MATHEMATISCHE ANNALEN, 388(4), 4201–4249. https://doi.org/10.1007/s00208-023-02609-7
Chicago author-date
Ghosh, Sekhar, Vishvesh Kumar, and Michael Ruzhansky. 2024. “Compact Embeddings, Eigenvalue Problems, and Subelliptic Brezis–Nirenberg Equations Involving Singularity on Stratified Lie Groups.” MATHEMATISCHE ANNALEN 388 (4): 4201–49. https://doi.org/10.1007/s00208-023-02609-7.
Chicago author-date (all authors)
Ghosh, Sekhar, Vishvesh Kumar, and Michael Ruzhansky. 2024. “Compact Embeddings, Eigenvalue Problems, and Subelliptic Brezis–Nirenberg Equations Involving Singularity on Stratified Lie Groups.” MATHEMATISCHE ANNALEN 388 (4): 4201–4249. doi:10.1007/s00208-023-02609-7.
Vancouver
1.
Ghosh S, Kumar V, Ruzhansky M. Compact embeddings, eigenvalue problems, and subelliptic Brezis–Nirenberg equations involving singularity on stratified Lie groups. MATHEMATISCHE ANNALEN. 2024;388(4):4201–49.
IEEE
[1]
S. Ghosh, V. Kumar, and M. Ruzhansky, “Compact embeddings, eigenvalue problems, and subelliptic Brezis–Nirenberg equations involving singularity on stratified Lie groups,” MATHEMATISCHE ANNALEN, vol. 388, no. 4, pp. 4201–4249, 2024.
@article{01H0XCTRRWH7RWHVC21AQTT5AN,
  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 G being the Heisenberg group.}},
  author       = {{Ghosh, Sekhar and Kumar, Vishvesh and Ruzhansky, Michael}},
  issn         = {{0025-5831}},
  journal      = {{MATHEMATISCHE ANNALEN}},
  keywords     = {{‘Ghent Analysis & PDE center,35R03,35H20,35P30,22E30,35R11,35J75,POSITIVE SOLUTIONS,DIFFERENTIAL-OPERATORS,ELLIPTIC-EQUATIONS,HARNACK INEQUALITY,DIRICHLET PROBLEM,CRITICAL GROWTH,SUB-LAPLACIANS,P-LAPLACIAN,EXISTENCE,MULTIPLICITY}},
  language     = {{eng}},
  number       = {{4}},
  pages        = {{4201--4249}},
  title        = {{Compact embeddings, eigenvalue problems, and subelliptic Brezis–Nirenberg equations involving singularity on stratified Lie groups}},
  url          = {{http://doi.org/10.1007/s00208-023-02609-7}},
  volume       = {{388}},
  year         = {{2024}},
}
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