A degree-counting formula for a Keller-Segel equation on a surface with boundary
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| Format: | Preprint |
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2025
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| author | Ahmedou, Mohameden Hu, Zhengni Wang, Heming |
| author_facet | Ahmedou, Mohameden Hu, Zhengni Wang, Heming |
| contents | In this paper, we consider the following Keller-Segel equation on a compact Riemann surface $(Σ, g)$ with smooth boundary $\partialΣ$: \[
-Δ_g u = ρ\Big(\frac{V e^u}{\int_Σ V e^u \mathrm{d} v_g} - \frac{1}{|Σ|_g}\Big) \text{ in } Σ, \quad \text{ with }
\partial_{ν_g} u = 0 \text{ on } \partial Σ, \]
where $V$ is a smooth positive function on $Σ$ and $ρ> 0$ is a parameter.
We perform a refined blow-up analysis of bubbling solutions and establish sharper a priori estimates around their concentration points. We then compute the Morse index of these solutions and use it to derive a counting formula for the Leray-Schauder degree in the non-resonant case (i.e., $ρ\notin 4 π\mathbb{N}$). Our approach follows the strategy suggested by Y. Y. Li [33] and later implemented by C.-S. Lin and C.-C. Chen [15,16] for the mean field equations on closed surfaces and employs techniques from Bahri's critical points at infinity [8]. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_12783 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A degree-counting formula for a Keller-Segel equation on a surface with boundary Ahmedou, Mohameden Hu, Zhengni Wang, Heming Analysis of PDEs 35J60, 35J25, 35R01 In this paper, we consider the following Keller-Segel equation on a compact Riemann surface $(Σ, g)$ with smooth boundary $\partialΣ$: \[ -Δ_g u = ρ\Big(\frac{V e^u}{\int_Σ V e^u \mathrm{d} v_g} - \frac{1}{|Σ|_g}\Big) \text{ in } Σ, \quad \text{ with } \partial_{ν_g} u = 0 \text{ on } \partial Σ, \] where $V$ is a smooth positive function on $Σ$ and $ρ> 0$ is a parameter. We perform a refined blow-up analysis of bubbling solutions and establish sharper a priori estimates around their concentration points. We then compute the Morse index of these solutions and use it to derive a counting formula for the Leray-Schauder degree in the non-resonant case (i.e., $ρ\notin 4 π\mathbb{N}$). Our approach follows the strategy suggested by Y. Y. Li [33] and later implemented by C.-S. Lin and C.-C. Chen [15,16] for the mean field equations on closed surfaces and employs techniques from Bahri's critical points at infinity [8]. |
| title | A degree-counting formula for a Keller-Segel equation on a surface with boundary |
| topic | Analysis of PDEs 35J60, 35J25, 35R01 |
| url | https://arxiv.org/abs/2506.12783 |