Is my solution to this exercise correct?
Consider the pde $e^xu_{xx}+e^yu_{yy}=u.$ Determine the region where the pde is elliptic, parabolic and hyperbolic.
My solution:
There's no region R where pde is parabolic since $e^{x+y}=0$ does not have solution.
Also there is no region R where pde is hyperbolic since $e^{x+y}>0,\forall (x,y)\in\mathbb R^2.$
There exists a region R when $x=0=y;x,y>0;x,y<0;x>0,y<0$ and $x<0,y>0,$ and in all this cases pde is elliptic.
(I'm going to write only one region since the rest of them are similar.)
If $x,y>0$,
$$R=\{(x,y)\in\mathbb R^2:x>0,y>0\}$$
$$e^xu_{xx}+e^yu_{yy}=u.$$ The canonical form is written like this $$A\partial_{xx} \psi+B\partial_{xy} \psi +C\partial_{yy}\psi+...=0$$ so $$\Delta=B^2-4AC=0-4e^xe^y=-4e^{x+y}$$
Elliptic for $\Delta<0$
Since $\Delta <0$ and $\forall (x,y) \in \mathbb{R^2}$ the equation is elliptic. So you are correct. But it's the case for all $(x,y) \in \mathbb{R^2}$