In my PDE lecture we had the following theorem and I am wondering how strong it is:
Theorem
Let $\Omega \subset \mathbb{R}^n$ be a bounded domain (with $\partial \Omega$ sufficiently smooth, lets say $\partial \Omega \subset C^\infty$) and $f \in C^1(\overline \Omega \times \mathbb{R})$. If there exists functions $\underline u, \overline u \in C^0(\overline \Omega) \cap C^2(\Omega)$ satisfying
$$ \begin{cases} -\Delta \underline{u} \le f(x, \underline{u}), & x \in \Omega, \\ \underline u \le 0, & x \in \partial \Omega \end{cases} $$
as well as
$$ \begin{cases} -\Delta \overline{u} \ge f(x, \overline{u}), & x \in \Omega, \\ \overline u \ge 0, & x \in \partial \Omega \end{cases} $$
and $\underline u \le \overline u$, then there exists a (classical) solution $u \in C^0(\overline \Omega) \cap C^2(\Omega)$ of
$$ \begin{cases} -\Delta u = f(x, u), & x \in \Omega, \\ u = 0, & x \in \partial \Omega. \end{cases} $$
satisfying $\underline u \le u \le \overline u$.
Question
Can I apply this theorem to prove the existence of a non-trivial solution where $f(x, u) = |u|^{p-1} u =: u^p$ for some $p \gt 1$?
I know for $\Omega = B_R(x_0), R \gt 0, x_0 \in \mathbb{R}^n$ there exists a positive solution, if $p \lt p_S := (\frac{n+2}{n-2})_+$. (I have seen a proof; that is definitely not trivial!)
So, for $p \lt p_S$ and $R \gt 0$ so large, that $\Omega \subset B_R(0)$ I have found a “super solution” $\overline u$ (which is “quite large”, if $R$ is “quite large”, ie. after finding some “sub solution” $\underline u$ the requirement $\underline u \le \overline u$ should not really be a problem.)
How to find a proper sub solution, if $\Omega$ is not a ball (by the way, that implies $n \ge 2$)? Is it even possible to prove the statement without knowing existence of a solution in the case of $\Omega = B_R$?
Note that $\underline u \equiv 0$ is a sub solution, but that does not help at all, as this only implies the existence of a solution $u \ge 0$ – which is trivial, as $u \equiv 0$ is a solution.