Blow-up solutions for the steady state of the Keller-Segel system on Riemann surfaces

We study the following Neumann boundary problem related to the stationary solutions of the Keller-Segel system, a basic model of chemotaxis phenomena: \ -g u + u = (Veᵘ_{ Veᵘ d vg}-1||g) in \ with ₆ u=0, on, where (, g) is a compact Riemann surface with the interior and the smooth boundary. Here, , 0 are non-negative parameters, and V is a smooth non-negative function with a finite zero set. For any given integers m k 0, we establish a sufficient condition on V for the existence of a sequence of blow-up solutions as approaches the critical values 4 (m+k). Moreover, the study expands to the corresponding singular problem.