  In this note, we study the curvature flow to Nirenberg problem on $S^2$ with
non-negative nonlinearity. This flow was introduced by Brendle and Struwe. Our
result is that the Nirenberg problems has a solution provided the prescribed
non-negative Gaussian curvature $f$ has its positive part, which possesses
non-degenerate critical points such that $\Delta_{S^2} f>0$ at the saddle
points.
