Remark on “A problem of prescribing Gaussian curvature on $S^2$" [\text{Proc. Amer. Math. Soc. 129 (2001), no. 12, 3757–3758}]

Abstract:

In this note, we remark on a 2001 paper of S. Goyal and V. Goyal. The main result of this work is that they used some elementary method to find a class of functions $K(x)=K(x_1,x_2,x_3)$ for which the solutions to \[ \Delta u + K(x)e^{2u} = 1\] on $S^2$ can be obtained. We observe that this class of functions that they studied is actually the trivial one, i.e. the class of positive constant functions.