Metrics of constant curvature on a Riemann surface with two corners on the boundary

In this paper we classify all solutions of{−Δu=|x|2αeu,in R+2,∂u∂t=c1|x|αeu2,on ∂R+2∩{s>0},∂u∂t=c2|x|αeu2,on ∂R+2∩{s<0} with the finite energy condition∫R+2|x|2αeudx<∞,∫∂R+2|x|αeu2ds<∞. Here c1c1, c2c2 are constants and α>−1α>−1.