I'm looking for someone who has read the article written by Brezis & Merle (https://doi.org/10.1080/03605309108820797). Because I'm confused when I saw the maximum principle. For example, See formula (3) in the proof of Theorem 1.
Brezis claimed that we only need $u\in L^{1}(\Omega), V(x)e^{u}\in L^{1}(\Omega)$ to ensure $-\Delta u=V(x)e^{u}$ make sense in the sense of distribution. When reading this article, I found he never uses the property of Sobolev space. So at first I believe in this paper, a solution $u(x)$ does not need to be in a Sobolev space.
But when I finish reading and review, everything fits the assumption except the maximum principle. I only learned that maximum principle when $u\in W^{1,2}(\Omega)$, and its proof can't extend to this article.
Question： Is that means Brezis achieved another maximum principle without existence of weak derivative? Or he indicated that we must consider solutions in $W^{1,2}(\Omega)$?
Maybe we can compare it to the equation$-\Delta u=\lambda \delta_{0}$ with or without boundary condition.