Brezis-Nirenberg type problem for fractional sub-Laplacian on the Heisenberg group
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| Format: | Preprint |
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2025
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| _version_ | 1866908496755163136 |
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| author | Naik, Vikram Yallapa Dwivedi, Gaurav |
| author_facet | Naik, Vikram Yallapa Dwivedi, Gaurav |
| contents | In this paper, we show the existence of a weak solution for a fractional sub-Laplace equation involving a term with the critical Sobolev exponent, namely, \begin{align*} (-Δ_\mathbb{H})^su - λu &= |u|^{Q^*_s -2}u \text{ in } Ω,\\ u &= 0 \text{ in } \mathbb{H}^N \setminus Ω, \end{align*} where $Ω\subseteq \mathbb{H}^N$ is bounded and has continuous boundary, $(-Δ_\mathbb{H})^s$ is the horizontal fractional Laplacian, $s \in (0,1), λ> 0,$ and $Q^*_s=\frac{2Q}{Q-2s}$ is the Sobolev critical exponent. This problem is motivated by the celebrated Brezis-Nirenberg problem \cite{brezis1983positive}. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_14990 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Brezis-Nirenberg type problem for fractional sub-Laplacian on the Heisenberg group Naik, Vikram Yallapa Dwivedi, Gaurav Analysis of PDEs 35R03, 35R11, 35H20 In this paper, we show the existence of a weak solution for a fractional sub-Laplace equation involving a term with the critical Sobolev exponent, namely, \begin{align*} (-Δ_\mathbb{H})^su - λu &= |u|^{Q^*_s -2}u \text{ in } Ω,\\ u &= 0 \text{ in } \mathbb{H}^N \setminus Ω, \end{align*} where $Ω\subseteq \mathbb{H}^N$ is bounded and has continuous boundary, $(-Δ_\mathbb{H})^s$ is the horizontal fractional Laplacian, $s \in (0,1), λ> 0,$ and $Q^*_s=\frac{2Q}{Q-2s}$ is the Sobolev critical exponent. This problem is motivated by the celebrated Brezis-Nirenberg problem \cite{brezis1983positive}. |
| title | Brezis-Nirenberg type problem for fractional sub-Laplacian on the Heisenberg group |
| topic | Analysis of PDEs 35R03, 35R11, 35H20 |
| url | https://arxiv.org/abs/2508.14990 |