Global branching for semilinear fractional Laplace with sublinear nonlinearity
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912701681238016 |
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| author | Abrantes, Jefferson Kumar, Rohit Sarkar, Abhishek |
| author_facet | Abrantes, Jefferson Kumar, Rohit Sarkar, Abhishek |
| contents | This article investigates the existence, nonexistence, and multiplicity of positive solutions to the sublinear fractional elliptic problem $(P_λ^s)$. We begin by establishing several a priori estimates that provide regularity results and describe the qualitative behavior of solutions. A critical threshold level for the parameter $λ$ is identified, which plays a crucial role in determining the existence or nonexistence of solutions. The sub and supersolution method is employed to obtain a weak solution. Furthermore, we establish a relation between the local minimizers of $\mathcal{D}^{s,2}(\mathbb{R}^N)$ versus $C(\mathbb{R}^N; 1+|x|^{N-2s})$. Combining these results with the Classical Linking Theorem, we demonstrate the existence of at least two distinct positive weak solutions to $(P_λ^s)$. This work extends the results of Yang, Abrantes, Ubilla, and Zhou (J. Differential Equations, 416:159-189, 2025) to the nonlocal setting, i.e., when $s \in (0,1)$. Several technical challenges arise in this framework, such as the lack of a standard comparison principle in $\mathbb{R}^N$ in the fractional setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_08037 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Global branching for semilinear fractional Laplace with sublinear nonlinearity Abrantes, Jefferson Kumar, Rohit Sarkar, Abhishek Analysis of PDEs 35A15, 35B38, 35B09, 35D30, 58E05, 35J61, 35R15 This article investigates the existence, nonexistence, and multiplicity of positive solutions to the sublinear fractional elliptic problem $(P_λ^s)$. We begin by establishing several a priori estimates that provide regularity results and describe the qualitative behavior of solutions. A critical threshold level for the parameter $λ$ is identified, which plays a crucial role in determining the existence or nonexistence of solutions. The sub and supersolution method is employed to obtain a weak solution. Furthermore, we establish a relation between the local minimizers of $\mathcal{D}^{s,2}(\mathbb{R}^N)$ versus $C(\mathbb{R}^N; 1+|x|^{N-2s})$. Combining these results with the Classical Linking Theorem, we demonstrate the existence of at least two distinct positive weak solutions to $(P_λ^s)$. This work extends the results of Yang, Abrantes, Ubilla, and Zhou (J. Differential Equations, 416:159-189, 2025) to the nonlocal setting, i.e., when $s \in (0,1)$. Several technical challenges arise in this framework, such as the lack of a standard comparison principle in $\mathbb{R}^N$ in the fractional setting. |
| title | Global branching for semilinear fractional Laplace with sublinear nonlinearity |
| topic | Analysis of PDEs 35A15, 35B38, 35B09, 35D30, 58E05, 35J61, 35R15 |
| url | https://arxiv.org/abs/2511.08037 |