Hello I have been fighting with this problems for a couples of days and cant get anything better, I will thanks a lot for your help, for little hint, some clue or idea :
Let $A$ be an Harmonic function and we know that $A\phi$ is harmonic.
And also we know that $A >0$
(Both function $A,\phi:R^2\rightarrow R$)
$\nabla ^2(\phi A) = A(\nabla ^2 \phi)+2\nabla A.\nabla \phi+\phi(\nabla ^2 A) $
and so: $0 = A(\nabla ^2 \phi)+2\nabla A.\nabla \phi$
With the Green identities we can say that:
$\int{A (\nabla\phi.n) dS }=-\int{ (\nabla A.\nabla \phi) dV } $
or
$\int{A (\nabla\phi.n) dS }=-\int{\phi (\nabla A.n) dS } $
Does that implies that $A$ is subharmonic or harmonic ?
Thanks!