Consider the PDE
P(x,y)∂2u∂x2+ex2ey2∂2u∂x∂y+Q(x,y)∂2u∂y2+e2x∂u∂x+ey∂u∂y=0
where P and Q are polynomials in two variables with real coefficients. Then which of the following is true for all choices of P and Q ?
1. There exists R>0 such that the PDE is elliptic in {(x,y)∈R2:x2+y2>R}
2. There exists R>0 such that the PDE is hyperbolic in {(x,y)∈R2:x2+y2>R}
3. There exists R>0 such that the PDE is parabolic in {(x,y)∈R2:x2+y2>R}
4. There exists R>0 such that the PDE is hyperbolic in {(x,y)∈R2:x2+y2<R}