So, I'm reading the classical paper


* Gidas, B., Ni, WM. & Nirenberg, L., *Symmetry and Related Properties via the Maximum Principle*, Commun. Math. Phys. **68** (1979) pp. 209–243, doi:[10.1007/BF01221125](https://doi.org/10.1007/BF01221125), [Project Euclid](https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-68/issue-3/Symmetry-and-related-properties-via-the-maximum-principle/cmp/1103905359.full),


and I am in trouble with some passages. Since giving a complete description of the framework where the problem is posed would be unfeasible, I seek help from people familiar with the referred paper, and this is why I'm posting it here instead of posting at MathStackExchange.


**First question**


In the proof of Lemma 2.1, when the hypothesis that $f(0) \geq 0$ in $\Omega\_\varepsilon$ is made, we obtain equation $\widehat{\text{(2.1)}}$:
$$
\Delta u + b\_1 u\_1 + f(u) - f(0) \leq 0.
$$
Then the authors claim that, by the Mean Value Theorem,
$$
\Delta u + b\_1 u\_1 + c(x) u \leq 0, \quad (\*)
$$
for some function $c(x)$.



> 
> How was the Mean Value Theorem used to yield $(\*)$?
> 
> 
> 


**Second question**


In the proof of Lemma 2.2, the authors claim that
$$
w(x) = v(x) - u(x) \leq 0, \quad w \not\equiv 0
$$
and
$$
\Delta w + b\_1(x) w\_1 + c(x) w \geq 0, \quad (\*\*)
$$
by the integral form of the Mean Value Theorem. Again,



> 
> How was the Mean Value Theorem used?
> 
> 
> 


**Third question**


Again in the proof of Lemma 2.2, the authors use the Maximum Principle for the equation $(\*\*)$



> 
> Do we know if c(x) is negative in order to apply the Maximum Principle? Is there a reference for a Maximum Principle where $c$ is anything (which is what they use, as I understand)?
> 
> 
> 


Thanks in advance.



Some questions on the famous Gidas-Ni-Nirenberg 1979 paper
For your first question: you just need that given a solution $u$ there exists a function $c(x)$ such that $f(u(x)) - f(0) = c(x) u(x)$. This is possible because by the Mean Value Theorem applied to $f$ there exists a function $v(x)$ between $u(x)$ and $0$ such that $f(u(x)) - f(0) = f'(v(x)) u(x)$.


For your second question, the MVT is applied to get $f(v(x)) - f(u(x)) = f'(z(x)) (v(x) - u(x))$ for some $z(x)$ between $v(x)$ and $u(x)$.


For your third question, the authors explain exactly this point on page 212 of the paper, in section 1.3. As they explained, the case $c(x) \leq 0$ is well-known, but the proof *for the specific statement that they need* also works for general-signed $c(x)$. They included a proof of the statement, which spans the bottom lines of page 212 and the top half of page 213.


