Nonlocal elliptic equations involving logarithmic Laplacian: Existence, non-existence and uniqueness results
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866908339044089856 |
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| author | Arora, Rakesh Giacomoni, Jacques Vaishnavi, Arshi |
| author_facet | Arora, Rakesh Giacomoni, Jacques Vaishnavi, Arshi |
| contents | In this work, we study the existence, non-existence, and uniqueness results for nonlocal elliptic equations involving logarithmic Laplacian, and subcritical, critical, and supercritical logarithmic nonlinearities. The Poho\u zaev's identity and Díaz-Saa type inequality are proved, which are of independent interest and can be applied to a larger class of problems. Depending upon the growth of nonlinearities and regularity of the weight function, we study the small-order asymptotic of nonlocal weighted elliptic equations involving the fractional Laplacian of order $2s.$ We show that the least energy solutions of a weighted nonlocal problem with superlinear or sublinear growth converge to a nontrivial nonnegative least-energy solution of Brézis-Nirenberg type and logistic-type limiting problem respectively involving the logarithmic Laplacian. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_15985 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Nonlocal elliptic equations involving logarithmic Laplacian: Existence, non-existence and uniqueness results Arora, Rakesh Giacomoni, Jacques Vaishnavi, Arshi Analysis of PDEs In this work, we study the existence, non-existence, and uniqueness results for nonlocal elliptic equations involving logarithmic Laplacian, and subcritical, critical, and supercritical logarithmic nonlinearities. The Poho\u zaev's identity and Díaz-Saa type inequality are proved, which are of independent interest and can be applied to a larger class of problems. Depending upon the growth of nonlinearities and regularity of the weight function, we study the small-order asymptotic of nonlocal weighted elliptic equations involving the fractional Laplacian of order $2s.$ We show that the least energy solutions of a weighted nonlocal problem with superlinear or sublinear growth converge to a nontrivial nonnegative least-energy solution of Brézis-Nirenberg type and logistic-type limiting problem respectively involving the logarithmic Laplacian. |
| title | Nonlocal elliptic equations involving logarithmic Laplacian: Existence, non-existence and uniqueness results |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2411.15985 |