# Is this PDE solvable?

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?