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, Project Euclid,

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.