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.

Lemma(Weak Comparison Principle). Suppose that in . Let satisfyand

for all non-negative . Then the inequality

on implies

in .

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,

and

Hence,

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.

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