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 that
on
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.