Quotient of solutions of a semilinear Dirichlet problem is $L^\infty$

Question

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

Answer 1

The fact that both $u_j$ solve the PDE is not an issue. What matters is that both $u_j$ are smooth over the closure $\overline\Omega$, positive in $\Omega$, vanish on $\partial\Omega$ and their normal derivatives don't vanish. This is where you may invoque L'Hopital's rule, if you like this tool.

What is more important is that we can with little effort derive an estimate of the ratio $w=\frac{u_2}{u_1}$. The paper assumes the important property that $u\mapsto f(x,u)$ is decreasing, which I shall use together with the maximum principle. This function satisfies the PDE $$-\Delta w-2\nabla\log u_1\cdot\nabla w=\frac1{u_1}(f(x,wu_1)-wf(x,u_1)).$$ If $w(\bar x)>1$ for some $\bar x\in\Omega$, then the assumption on $f$ ensures that $f(x,wu_1)-wf(x,u_1)\le0$ and thus $-\Delta w-2\nabla\log u_1\cdot\nabla w\le0$. By the maximum principle, $w$ cannot achieve a local maximum at $\bar x$. This shows that either $w\le1$ everywhere, or it achieves its maximum at the boundary. In the latter case, $w$ coincides (L'Hopital) with the ratio of normal derivatives. In conclusion, we have $$\max_\Omega\frac{u_2}{u_1}\le\max\left\{1,\max_{\partial\Omega}\frac{\partial u_2/\partial\nu}{\partial u_1/\partial\nu}\right\}.$$