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

TL;DR

This paper analyzes solutions to a Keller-Segel equation on a surface with boundary, providing a degree-counting formula by studying solution blow-up behavior, Morse index, and a priori estimates.

Contribution

It introduces a refined blow-up analysis and computes the Morse index to establish a new degree-counting formula for solutions on surfaces with boundary.

Findings

Sharper a priori estimates around concentration points

Computed Morse index of bubbling solutions

Derived a degree-counting formula for the Leray-Schauder degree

Abstract

In this paper, we consider the following Keller-Segel equation on a compact Riemann surface $(\Sigma, g)$ with smooth boundary $\partial\Sigma$: \[ -\Delta_g u = \rho\Big(\frac{V e^u}{\int_{\Sigma} V e^u \mathrm{d} v_g} - \frac{1}{|\Sigma|_g}\Big) \text{ in } {\Sigma}, \quad \text{ with } \partial_{\nu_g} u = 0 \text{ on } \partial \Sigma, \] where $V$ is a smooth positive function on $\Sigma$ 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., $\rho \notin 4 \pi \mathbb{N}$). 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].