Quantitative properties of the Hardy-type mean field equation
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915787216781312 |
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| author | Chen, Lu Wang, Bohan Wang, Chunhua |
| author_facet | Chen, Lu Wang, Bohan Wang, Chunhua |
| contents | In this paper, we consider the following Hardy-type mean field equation \[ \left\{ {\begin{array}{*{20}{c}} { - Δu-\frac{1}{(1-|x|^2)^2} u = λe^u}, & {\rm in} \ \ B_1,\\ {\ \ \ \ u = 0,} &\ {\rm on}\ \partial B_1, \end{array}} \right. \] \[\] where $λ>0$ is small and $B_1$ is the standard unit disc of $\mathbb{R}^2$. Applying the moving plane method of hyperbolic space and the accurate expansion of heat kernel on hyperbolic space, we establish the radial symmetry and Brezis-Merle lemma for solutions of Hardy-type mean field equation. Meanwhile, we also derive the quantitative results for solutions of Hardy-type mean field equation, which improves significantly the compactness results for classical mean-field equation obtained by Brezis-Merle and Li-Shafrir. Furthermore, applying the local Pohozaev identity from scaling, blow-up analysis and a contradiction argument, we prove that the solutions are unique when $λ$ is sufficiently close to 0. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_17886 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quantitative properties of the Hardy-type mean field equation Chen, Lu Wang, Bohan Wang, Chunhua Analysis of PDEs In this paper, we consider the following Hardy-type mean field equation \[ \left\{ {\begin{array}{*{20}{c}} { - Δu-\frac{1}{(1-|x|^2)^2} u = λe^u}, & {\rm in} \ \ B_1,\\ {\ \ \ \ u = 0,} &\ {\rm on}\ \partial B_1, \end{array}} \right. \] \[\] where $λ>0$ is small and $B_1$ is the standard unit disc of $\mathbb{R}^2$. Applying the moving plane method of hyperbolic space and the accurate expansion of heat kernel on hyperbolic space, we establish the radial symmetry and Brezis-Merle lemma for solutions of Hardy-type mean field equation. Meanwhile, we also derive the quantitative results for solutions of Hardy-type mean field equation, which improves significantly the compactness results for classical mean-field equation obtained by Brezis-Merle and Li-Shafrir. Furthermore, applying the local Pohozaev identity from scaling, blow-up analysis and a contradiction argument, we prove that the solutions are unique when $λ$ is sufficiently close to 0. |
| title | Quantitative properties of the Hardy-type mean field equation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2412.17886 |