Let $M = mathbb{R}^3 setminus B_1$ where $B_1$ is unit ball. I am trying to solve the following PDE for $f$:

$$Delta f -frac{ f }{r^2}+ frac{ left. f right|_{partial M}}{r^2} = 0, qquad text{on} , M$$

$$f + a partial_r f = h, qquad text{on} , partial M$$

$$lim_{rto infty} f= 0 $$

where $a>0$ is a constant and $h in L^2(partial M)$ (or any nice function space you want).

Note that this PDE has a nonlocal part, namely $left. f right|_{partial M} $, and so it’s not really a quasilinear PDE.

Also note that it is easy to see right away that $sup_{x in M} f(x) leq max{sup_{xin partial M}f,0} $. This can be proven by contradiction; if the maximum is in the interior, then $Delta f<0$ implying $f – left. f right|_{partial M} <0$, which is a contradiction.

How do I prove existence and uniqueness? Can Perron method be applied?