Nonlocal singular problem and associated Sobolev type inequality with extremal in the Heisenberg group
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866916921118556160 |
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| author | Garain, Prashanta |
| author_facet | Garain, Prashanta |
| contents | We study a fractional $p$-Laplace equation involving a variable exponent singular nonlinearity in the framework of the Heisenberg group. We first establish the existence and regularity of weak solutions. In the case of a constant singular exponent, we further prove the uniqueness of solutions and characterize the extremals of a related Sobolev-type inequality. Additionally, we demonstrate a connection between the solutions of the singular problem and these extremals. To the best of our knowledge, these findings provide new insights even in the classical case $p=2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_19811 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Nonlocal singular problem and associated Sobolev type inequality with extremal in the Heisenberg group Garain, Prashanta Analysis of PDEs 35J62, 35J75, 35R03, 35R11, 35A02, 35A23, 35B45 We study a fractional $p$-Laplace equation involving a variable exponent singular nonlinearity in the framework of the Heisenberg group. We first establish the existence and regularity of weak solutions. In the case of a constant singular exponent, we further prove the uniqueness of solutions and characterize the extremals of a related Sobolev-type inequality. Additionally, we demonstrate a connection between the solutions of the singular problem and these extremals. To the best of our knowledge, these findings provide new insights even in the classical case $p=2$. |
| title | Nonlocal singular problem and associated Sobolev type inequality with extremal in the Heisenberg group |
| topic | Analysis of PDEs 35J62, 35J75, 35R03, 35R11, 35A02, 35A23, 35B45 |
| url | https://arxiv.org/abs/2508.19811 |