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: \[ -Δ_g u = ρ(V e^u/∫_Σ V e^u d v_g - 1/|Σ|_g) in Σ, with ∂ν_g u = 0 on ∂ Σ, \] 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].