I would appreciate help in completing the proof of the equation on the
second line of the attached file.
What you need to prove is not that div[phi E] = 0, but rather that div[phi E] integrated over a volume approaches zero in the limit that the volume becomes infinitely large. If div[phi E] were a positive definite quantity then this would also imply that div[phi E] = 0 everywhere, but it isn't! div[phi E] can be negative or positive.
I'm now going to derive the result step by step.
Consider the volume integral:
Volume integral [rho phi].
Replacing rho = epsilon_0 div E ...
The electrostatic field and potentials are determined. A detailed solution is given.