A best constant and the Gaussian curvature

For axisymmetric $f \in {C^\infty }({S^2})$ we find conditions to make $f$ the scalar curvature of a metric pointwise conformal to the standard metric of ${S^2}$. Closely related to these results, we prove that in the inequality (Moser [8])

$$\int_{{S^2}} {{e^u} \leq C{e^{\left\Vert {\nabla u} \right\Vert _... ...16\pi \quad }}\forall u \in H_1^2({S^2})} {\text{ with }}\int_{{S^2}} {u = 0} ,$$

, the best constant $C = {\text{Vol(}}{{\text{S}}^2}{\text{)}}$.