Year: 2026
Type: article
Abstract: 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 Li [Harnack type inequality: The method of moving planes, Comm. Math. Phys. 200(2) (1999) 421–444] and later implemented by Chen and Lin [Sharp estimates for solutions of multi-bubbles in compact Riemann surfaces, Comm. Pure Appl. Math. 55(6) (2002) 728–771; Topological degree for a mean field equation on Riemann surfaces, Comm. Pure Appl. Math. 56(12) (2003) 1667–1727] for the mean field equations on closed surfaces and employs techniques from Bahri’s critical points at infinity [Critical Points Theory at Infinity in Some Variational Problems, Pitman Research Notes in Mathematics Series, Vol. 182 (Longman Scientific & Technical, New York, 1989)].
Authors: Mohameden Ahmedou, Zhengni Hu, Heming Wang
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4415112499