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 [Formula: see text] with smooth boundary [Formula: see text]: [Formula: see text] where [Formula: see text] is a smooth positive function on [Formula: see text] and [Formula: see text] 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., [Formula: see text]). 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].