A degree-counting formula for a Keller-Segel equation on a surface with boundary

In this paper, we consider the following Keller-Segel equation on a compact Riemann surface $(\mathrm{\Sigma },g)$ with smooth boundary $\mathrm{\partial }\mathrm{\Sigma }$: $$-{\mathrm{\Delta }}_{g}u=\rho (\frac{V{e}^u}{{\int }_{\mathrm{\Sigma }}V{e}^u\mathrm{d}{v}_g}-\frac{1}{\mathrm{\mid }\mathrm{\Sigma }{\mathrm{\mid }}_g})\text{ in }\mathrm{\Sigma },\text{ with }{\mathrm{\partial }}_{{\nu }_g}u=0\text{ on }\mathrm{\partial }\mathrm{\Sigma },$$ where $V$ is a smooth positive function on $\mathrm{\Sigma }$ and $\rho >0$ 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., $\rho \mathrm{\notin }4\pi \mathbb{N}$). 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].