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.
Lemma (Weak Comparison Principle). Suppose that in . Let satisfy
for all non-negative . Then the inequality
A proof for the weak comparison principle is rather standard as follows: Thanks to the condition on the boundary , we can use as a test function. Clearly,
due to the monotonicity of . Hence, the proof follows.
The second auxilary result is the so-called Hopf Maximum Principle as the following
Lemma (Hopf’s Maximum Principle). Let is a geodesic ball contained in and assume that in and for some point . Furthermore, we assume that all assumptions in the Maximum Principle are fulfilled. Then
To prove, W.L.O.G, we may assume that is the unit geodesic ball centered in in . Then we define
for some and .
An elementary calculation shows that there is an satisfying
in independent of . Now, we choose in such a way that
on . The Weak Comparison Principle then implies that
in . Then the Lemma follows.
We are now in a position to prove our main theorem.
Proof of Theorem. We prove the theorem by contradiction. Indeed, suppose that there is a point such that . Then, we can find a small ball in such a way that and in .
Hence, is a minimum point of in , this implies that . However, by the Hopf Maximum Principle above, it should be , a contradiction.
We now consider a simple example arising from the prescribed scalar curvature equation given by
for smooth functions and . Suppose that is a non-negative of the equation, we show that either or . To do so, we simply write
where is a constant chosen in such a way that
for all . Hence, there holds
pointwise in . It is immediately to see that our function fulfills all conditions of the Maximum Principle (monotonicity + growth), hence either or as claimed.
It is important to note that the advantage of the above result is that has no fix sign which could be sign-changing.