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## reference request – Some questions on the well-known Gidas-Ni-Nirenberg 1979 paper

So, I’m studying the classical paper *residue and Related Properties through the Maximum Principle*, by Gidas, Ni and Nirenberg, and I’m in wretchedness with some passages. Since giving an entire description of the framework the place the issue is posed would breathe unfeasible, I hunt assist from folks immediate with the referred paper, and this is the reason I’m posting it right here as a substitute of posting at MathStackExchange.

**First query**

In the proof of Lemma 2.1, when the speculation that $f(0) geq 0$ in $Omega_varepsilon$ is made, we safe equation $widehat{textual content{(2.1)}}$:

$$

Delta u + b_1 u_1 + f(u) – f(0) leq 0.

$$

Then the authors pretense that, by the denote Value Theorem,

$$

Delta u + b_1 u_1 + c(x) u leq 0, quad (*)

$$

for some obligation $c(x)$.

How was the denote Value Theorem used to relent $(*)$?

**Second query**

In the proof of Lemma 2.2, the authors pretense that

$$

w(x) = v(x) – u(x) leq 0, quad w notequiv 0

$$

and

$$

Delta w + b_1(x) w_1 + c(x) w geq 0, quad (**)

$$

by the integral design of the denote Value Theorem. Again,

How was the denote Value Theorem used?

**Third query**

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

assassinate we all know if c(x) is traverse in bid to use the Maximum Principle? Is there a reference for a Maximum Principle the place $c$ is something (which is what they make use of, as I win)?

Thanks in close by.

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