Uniqueness and multiplicity of positive radial solutions to the super-critical Brezis-Nirenberg problem in an annulus
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866913931246698496 |
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| author | Shioji, Naoki Tanaka, Satoshi Watanabe, Kohtaro |
| author_facet | Shioji, Naoki Tanaka, Satoshi Watanabe, Kohtaro |
| contents | The super-critical Brezis-Nirenberg problem in an annulus is considered. The new uniqueness result of positive radial solutions is established for the three-dimensional case. It is also proved that the problem has at least three positive radial solutions when the inner radius of the annulus is sufficiently small and the outer radius of the annulus is in a certain range. Moreover, for each positive integer $k$, the problem has at least $k$ positive radial solutions when the exponent of the equation is greater than the critical Sobolev exponent and is less than the Joseph-Lundgren exponent. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_15220 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Uniqueness and multiplicity of positive radial solutions to the super-critical Brezis-Nirenberg problem in an annulus Shioji, Naoki Tanaka, Satoshi Watanabe, Kohtaro Analysis of PDEs 35J61, 35A24, 35J25 The super-critical Brezis-Nirenberg problem in an annulus is considered. The new uniqueness result of positive radial solutions is established for the three-dimensional case. It is also proved that the problem has at least three positive radial solutions when the inner radius of the annulus is sufficiently small and the outer radius of the annulus is in a certain range. Moreover, for each positive integer $k$, the problem has at least $k$ positive radial solutions when the exponent of the equation is greater than the critical Sobolev exponent and is less than the Joseph-Lundgren exponent. |
| title | Uniqueness and multiplicity of positive radial solutions to the super-critical Brezis-Nirenberg problem in an annulus |
| topic | Analysis of PDEs 35J61, 35A24, 35J25 |
| url | https://arxiv.org/abs/2501.15220 |