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: \[ -Δ_g u +β u =λ({Ve^u}{∫Σ Ve^u d v_g}-1/|Σ|_g) in Σ \] with ∂νg u=0, on ∂ Σ, where (Σ, g) is a compact Riemann surface with the interior Σ and the smooth boundary ∂ Σ. Here, λ, β≥ 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 λ approaches the critical values 4π(m+k). Moreover, the study expands to the corresponding singular problem.