I try to find some references about the theory of elliptic PDE in an infinite domain. The references I am aware of (Brezis, Evans) present only the case where the equation is posed in bounded open subset of $\mathbb{R}^n$.
Specifically I consider the following PDE in $\mathbb{R}^n$:
$$ A(x,y)^2u_{xx}+B(x,y)^2u_{yy}+C(x,y)u_x+D(x,y)u_y=0 $$
where $A,B,C$ and $D$ are $C^{\infty}$ smooth.
Question: When does a solution exists (I don't care of uniqueness) ? When is this solution $C^{\infty}$ smooth ?