Profile of blow-up solutions to the exponential Neumann boundary value problem

Abstract

Arising in problems of prescribed Gaussian and Geodesic curvatures, we consider a blow-up sequence of solutions to the following exponential Neumann boundary value problem: −Δun=Vn(x)|x|2αne2uninBR+,∂un∂ν=hn(x)|x|αneunon∂BR+⋂∂R+2,where ν denotes the outer unit normal vector to ∂BR+⋂∂R+2 and αn→α∈(−1,+∞). We establish a uniform estimate for this blow-up sequence of solutions near an isolated singular blow up point. We treat two cases α∉N and α∈N+ separately, and obtain two different uniform pointwise estimates.