Blow-up solutions for mean field equations with Neumann boundary conditions on Riemann surfaces
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916550516146176 |
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| author | Hu, Zhengni Bartsch, Thomas Ahmedou, Mohameden |
| author_facet | Hu, Zhengni Bartsch, Thomas Ahmedou, Mohameden |
| contents | On a compact Riemann surface $(Σ, g)$ with a smooth boundary $\partial Σ$, we consider the following mean field equations with Neumann boundary conditions: $$ -Δ_g u = λ\left(\frac{Ve^u}{\int_Σ Ve^u \, dv_g} - \frac{1}{|Σ|_g}\right) \text{ in } Σ\text{ with } \partial_{ν_g} u = 0 \text{ on } \partial Σ, $$ We find conditions on the potential function $V: Σ\to \mathbb{R}^+$ such that solutions exist for the parameter $λ$ when it is in a small right (or left) neighborhood of a critical value $4π(m+k)$ for $k \leq m \in \mathbb{N}_+$ and blow up as $λ$ approaches the critical parameter. The blow-up occurs exactly at $k$ points in the interior of $Σ$ and $(m-k)$ points on the boundary $\partial Σ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_16917 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Blow-up solutions for mean field equations with Neumann boundary conditions on Riemann surfaces Hu, Zhengni Bartsch, Thomas Ahmedou, Mohameden Analysis of PDEs 35B33 (Primary) 35J61, 35R01 (Secondary) On a compact Riemann surface $(Σ, g)$ with a smooth boundary $\partial Σ$, we consider the following mean field equations with Neumann boundary conditions: $$ -Δ_g u = λ\left(\frac{Ve^u}{\int_Σ Ve^u \, dv_g} - \frac{1}{|Σ|_g}\right) \text{ in } Σ\text{ with } \partial_{ν_g} u = 0 \text{ on } \partial Σ, $$ We find conditions on the potential function $V: Σ\to \mathbb{R}^+$ such that solutions exist for the parameter $λ$ when it is in a small right (or left) neighborhood of a critical value $4π(m+k)$ for $k \leq m \in \mathbb{N}_+$ and blow up as $λ$ approaches the critical parameter. The blow-up occurs exactly at $k$ points in the interior of $Σ$ and $(m-k)$ points on the boundary $\partial Σ$. |
| title | Blow-up solutions for mean field equations with Neumann boundary conditions on Riemann surfaces |
| topic | Analysis of PDEs 35B33 (Primary) 35J61, 35R01 (Secondary) |
| url | https://arxiv.org/abs/2408.16917 |