Harmonic function product, Knowing that one is Harmonic implies something about the other? 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!
 A: Another way to phrase the question: what can we say about the ratio of two harmonic functions? I'll use different notation: $f =u/v$ where $u$ and $v$ are harmonic. Then:   


*

*$f$ is real-analytic in its domain of definition (where $  v\ne 0$)

*Every level set of $f$ is the zero set of a harmonic function ($f=c$ means $u-cv=0$). In particular, a level set of $f$ cannot contain a closed curve (unless the domain of definition of $u,v$ is multiply connected). Zero sets of harmonic function are quite special: for example, they cannot contain a piece of cubic curve such as $y=x^3$. See this paper for more. 

*$f$ has no local extrema unless it is constant. Again, this follows by applying the maximum principle to $u-cv$.


On the other hand, $f$ need not be harmonic or sub- or super- harmonic. A simple example is $1/x$ which is subharmonic in the right halfplane and superharmonic in the other. It is less trivial to find an example where $\Delta f$ changes sign within a connected component of the domain of $f$. Here is one: $f(x,y)=\dfrac{xy}{x^2-y^2}$ has $\Delta f = \dfrac{8xy(x^2+y^2)}{(x^2-y^2)^3}$. 
