I posted this on MathStackExchange, but it hasn't even got 10 views, so probably it is better to post here. I hope it is not inappropriate.

I am reading a paper of Brezis and Oswald about existence and uniqueness of positive solutions to sublinear elliptic equations: \begin{equation} - \Delta u = f(x, u), \ u \geq 0, \ u \not\equiv 0 \text{ in } \Omega, \qquad u = 0 \text{ on } \partial \Omega. \end{equation} As usual, $\Omega$ is a bounded smooth domain of $\mathbb R^N$ and there are certain hypotheses on $f$, but I think they are not important for my question.

They prove in the paper that any solution of the problem (given the suitable hypotheses on $f$) satisfy $$ (*) \qquad u > 0 \text{ in } \Omega, \qquad \frac{\partial u}{\partial \nu} < 0 \text{ on } \partial \Omega, $$ where $\nu$ is the outer unit normal. Then they claim that from $(*)$ it follows that if $u_1, u_2$ are two solutions of the problem, then $u_1/u_2 \in L^\infty(\Omega)$. In the book "Qualitative Analysis of Nonlinear Elliptic Partial Differential Equations" Radulescu also presents this result and explains that it follows by L'Hospital's rule, but I also don't understand this.

I) Why does $u_1/u_2 \in L^\infty(\Omega)$?

II) Does I) still hold if $u$ is a solution of the problem $$ - \Delta u = f(x, u) \text{ in } \Omega, \qquad u = 0 \text{ on } \Gamma, \qquad \frac{\partial u}{\partial \nu} = 0 \text{ on } \Gamma_1, $$ where $\partial \Omega = \Gamma \cup \Gamma_1$, both parts being smooth? It is known that $(*)$ still holds in this case with $\Gamma$ in the place of $\partial \Omega$.

The paper: https://sites.math.rutgers.edu/~brezis/PUBlications/110-journal.pdf