Consider the manifold $M = \mathbb{R}^3 \setminus B$ where $B$ is the ball with radius $1$ with riemannian metric $g$ (not necessarily the euclidean metric).
I am looking for solutions to $\Delta_g f = 0$ where $f \to 1 $ at infinity and $f=f_0$ on $\partial M$ where $f_0$ is some positive function on $\partial M$
What can we say about existence and uniqueness of this PDE? Any simple proofs?
What can we say about the asymptotic behaviour of $f$ at infinity?
I think if g is the euclidean metric, the asymptotics are: $$f = 1 + \frac{C}r + O(\frac{1}{r^2}) $$ where $C$ is some constant and $r = \sqrt{x^2 + y^2 + z^2}$ in cartesian coordinates.
What if $g$ is asymptotically flat? (so in cartesian coordinates, the metric satisfies $g_{ij} = \delta_{ij} + O \left(\frac{1}{r^{\delta}}\right) $ for some $\delta > 0 $).
Any help is appreciated. If you know any references, please share it with me.