Don't understand a PDE argument involving $L^p$ norms and inequalities

Question

I'm reading this paper. I do not understand how the author proves Corollary 5.12 for the case $p=2$. He addresses everything except $p=2$ but claims it holds for all $p$. Can someone help me to see why it is true?

He uses Proposition 5.9, which I quote here.

Here is the weak form of the equation. Choosing the test function to be $u$, I almost get the identity (5.36) for $p=2$ but the $\delta$ term is wrong.