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: $$\{\begin{array}{ll}-{\mathrm{\Delta }}_{g}u+\beta u=\lambda (\frac{V{e}^u}{{\int }_{\mathrm{\Sigma }}V{e}^{u}d{v}_g}-1),& \text{ in }\mathring{\mathrm{\Sigma }}\\ {\mathrm{\partial }}_{{\nu }_g}u=0,& \text{ on }\mathrm{\partial }\mathrm{\Sigma }\end{array},$$ on a compact Riemann surface $(\mathrm{\Sigma },g)$ of unit area, with interior $\mathring{\mathrm{\Sigma }}$ and smooth boundary $\mathrm{\partial }\mathrm{\Sigma }$. Here, ${\mathrm{\Delta }}_g$ denote the Laplace-Beltrami operator, $d{v}_g$ the area element of $(\mathrm{\Sigma },g)$, and ${\nu }_g$ the unit outward normal to $\mathrm{\partial }\mathrm{\Sigma }$ and $\lambda$ and $\beta$ are non-negative parameters, $V$ is non-negative with finite zero set. For any integers $m>0$ and $k,l\ge 0$ 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.