I'm reading this paper https://arxiv.org/abs/1401.7366 and trying to prove Corollary 4.6.
The result essentially says the following. Let $B_r \subseteq \mathbb R^4$ be the ball of radius $r$ with the standard metric. Suppose $\phi: B_1 \to \mathbb R$, ($\phi$ smooth and bounded) satisfies the equation $$ \Delta |\phi|^2 = Q(\phi, \phi) + P(\phi). $$ Here $Q(\phi, \phi)$ is some quadratic function where we have a bound $$ |Q(\phi, \phi)| \leq C|\phi|^2 $$ for some $C$, and $P(\phi) \geq 0$. With this setup they claim that there is a constant $D$, independent of $\phi$, such that $$ \sup_{x \in B_{1 - \varepsilon}}|\phi|^2 \leq D\int_{B_1} |\phi|^2. $$
Apparently this is just an application of the maximum principle. By this I think that the proof should go in the following way. Let $u$ be a solution to Poisson's equation
$$ \Delta u = Q(\phi, \phi), \quad u|_{\partial B_1} = |\phi|^2. $$
Then the maximum principle applied to $|\phi|^2 - u$ gives that $|\phi|^2 \leq u$. Now I think there should be some facts about solutions to Poisson's equation that give estimates of the form
$$ \sup_{x \in B_{1 - \varepsilon}} u \leq A\|\Delta u\|_{L^1(B)} $$
from which the result would follow, but I'm unsure of how to prove such an estimate.