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 (Σ, g) with smooth boundary Σ: \ -Δg u = ρ (V eᵘ_Σ V eᵘ d vg - 1|Σ|g) in Σ, with ⏜₆ u = 0 on 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 4 πN). 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.