Consider $P(x,y)$ and $Q(x,y)$ two $C^2$ real functions such that their Laplacian operators are non-zero everywhere in their domaisn, that is
$\frac{\partial^2 P}{\partial x^2}+\frac{\partial^2 P}{\partial y^2}\neq0$
$\frac{\partial^2 Q}{\partial x^2}+\frac{\partial^2 Q}{\partial y^2}\neq0$
What are the minimum requirements that $P, Q$ must have in order to a map $F=(P,Q)$ be injective?