Finding a solutions for an equation

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

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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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