Existence of Positive Solutions for Generalized Fractional Brézis-Nirenberg Problem
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| Format: | Preprint |
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2024
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| _version_ | 1866911911541473280 |
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| author | Kumar, Rohit Sarkar, Abhishek |
| author_facet | Kumar, Rohit Sarkar, Abhishek |
| contents | In this article, we study the fractional Brézis-Nirenberg type problem on whole domain $\mathbb{R}^N$ associated with the fractional $p$-Laplace operator. To be precise, we want to study the following problem: \begin{equation*}
(-Δ)_{p}^{s}u - λw |u|^{p-2}u= |u|^{p_{s}^{*}-2}u \quad \text{in} ~\mathcal{D}^{s,p}(\mathbb{R}^{N}), \end{equation*} where $s\in (0,1),~p \in (1,\frac{N}{s}), ~p_{s}^{*}= \frac{Np}{N-sp}$ and the operator $(-Δ)_{p}^{s}$ is the fractional $p$-Laplace operator. The space $\mathcal{D}^{s,p}(\mathbb{R}^{N})$ is the completion of $C_c^\infty(\mathbb{R}^N)$ with respect to the Gaglairdo semi-norm. In this article, we prove the existence of a positive solution to this problem by allowing the Hardy weight $w$ to change its sign. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_05793 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Existence of Positive Solutions for Generalized Fractional Brézis-Nirenberg Problem Kumar, Rohit Sarkar, Abhishek Analysis of PDEs 35B09, 35R11 In this article, we study the fractional Brézis-Nirenberg type problem on whole domain $\mathbb{R}^N$ associated with the fractional $p$-Laplace operator. To be precise, we want to study the following problem: \begin{equation*} (-Δ)_{p}^{s}u - λw |u|^{p-2}u= |u|^{p_{s}^{*}-2}u \quad \text{in} ~\mathcal{D}^{s,p}(\mathbb{R}^{N}), \end{equation*} where $s\in (0,1),~p \in (1,\frac{N}{s}), ~p_{s}^{*}= \frac{Np}{N-sp}$ and the operator $(-Δ)_{p}^{s}$ is the fractional $p$-Laplace operator. The space $\mathcal{D}^{s,p}(\mathbb{R}^{N})$ is the completion of $C_c^\infty(\mathbb{R}^N)$ with respect to the Gaglairdo semi-norm. In this article, we prove the existence of a positive solution to this problem by allowing the Hardy weight $w$ to change its sign. |
| title | Existence of Positive Solutions for Generalized Fractional Brézis-Nirenberg Problem |
| topic | Analysis of PDEs 35B09, 35R11 |
| url | https://arxiv.org/abs/2406.05793 |