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

Abstract

We study the following Neumann boundary problem related to the stationary solutions of the Keller-Segel system, a basic model of chemotaxis phenomena: \[ \left\{\begin{array}{ll} -\Delta_g u +\beta u =\lambda\left(\frac{Ve^u}{\int_{\Sigma} Ve^u d v_g}-1\right), &\text { in } \mathring\Sigma\\ \partial_{ \nu_g} u=0, &\text { on } \partial \Sigma \end{array} \right.,\] on a compact Riemann surface $(\Sigma, g)$ of unit area, with interior $\mathring\Sigma$ and smooth boundary $\partial \Sigma$. Here, $\Delta_g$ denote the Laplace-Beltrami operator, $dv_g$ the area element of $(\Sigma, g)$, and $\nu_g$ the unit outward normal to $\partial \Sigma$ and $\lambda$ and $\beta$ are non-negative parameters, $V$ is non-negative with finite zero set. For any integers $m>0$ and $k,l\geq 0$ with $m=2k+l$, we establish a sufficient condition on $V$ for the existence of a sequence of blow-up solutions as $\lambda$ approaches the critical values $4\pi m$, which blows up at $k$ points in the interior and $l$ points on the boundary. Moreover, the study expands to the corresponding singular problem.