I'm reading through Partial Differential Equations by Lawrence C. Evans and am having an issue understanding a part of a proof regarding Hopf's Lemma.
The version of Hopf's lemma proved in 6.4.2 states that for a linear eliptical PDE with the following form (and some conditions on $(a_{ij})$:
$$\sum_{i,j=1}^n a_{ij}(x)u_{{x_i}{x_j}}(x) + \sum_{i}^n b_{i}(x)u_{x_i}(x) + c(x)u_{{x_i}{x_j}}(x) = f(x) $$
where everything followed by (x) is a function from $U$ to $\mathbb{R}$.
Then in a much later proof in chapter 9.5.2 he starts with a function $u$ of satisfying
$$-\triangle u = f(u) \text{ in }U$$ $$ u \equiv 0 \text{ on } \partial U$$
for $f: \mathbb{R} \to \mathbb{R}$ with $f(0) > 0$ and in order to get to the form above he writes (for the interior of $U$)
$$-\triangle u - f(u) = -\triangle u - f(u) + f(0) - f(0) \leq -\triangle u - (f(u) - f(0))$$
and claims we can write
$$f(u) - f(0) = \int_0^1f'(su(x))ds \cdot u = c(x) \cdot u$$
where $c(x)$ doesn't depend on $u$ and we can therefore use Hopf's lemma defined above.
I understand that
$$\int_0^{u(x)} f(s)ds = \frac{1}{u(x)}\int_0^1f'(su(x))ds$$
and therefore the form is valid, what I do not understand is why can he claim that this function $c$ does not depend on $u$, and therefore the PDE can be regarded as linear. I think it pretty obviously changes when the function $u$ changes. As I understand this logic I can also claim that $-\triangle u = u^2$ is linear as I can write $u^2(x) = c(x)u(x)$ for $c(x)=u(x)$..
I've discussed this with my Professor, and apparently even though $c$ is assumed to be linear in the relevant chapter, the proof of the Lemma never uses the linearity of $c$, and it therefore can be applied also when $c$ is not linear.