In this note, we consider a strong maximum principle due to Tolksdorf and application to the prescribed scalar curvature equation. In the context of closed and compact Riemannian manifolds of dimension , it can be stated as follows

Theorem(Maximum Principle). Let be a compact Riemannian manifold of dimension . Let be such thaton where is a function such that

(i.e. is monotone increasing w.r.t the variable ) and that

for all and for some constant . If in and does not vanish identically, then in .

To prove this theorem, we need two auxiliary results. The first result is the weak comparison principle which can be stated as the following.