Asymptotical Behaviors for Neumann Boundary Problem with Singular Data

In this paper, we will analyze the blow-up behaviors for solutions to the Laplacian equation with exponential Neumann boundary condition. In particular, the boundary value is with a kind of singular data. We show a Brezis–Merle type concentration-compactness theorem, calculate the blow up value at the blow-up point, and give a point-wise estimate for the profile of the solution sequence at the blow-up point.