Liouville type equation with exponential Neumann boundary condition and with singular data

In this paper we will analyze the blow-up behaviors of solutions to the singular Liouville type equation with exponential Neumann boundary condition. We generalize the Brezis–Merle type concentration-compactness theorem to this Neumann problem. Then along the line of the Li–Shafrir type quantization property we show that the blow-up value (m(0) in 2pi mathbb Ncup { 2pi (1+alpha )+2pi (mathbb Ncup {0})}) if the singular point 0 is a blow-up point. In the end, when the boundary value of solutions has an additional condition, we can obtain the precise blow-up value (m(0)=2pi (1+alpha )).