Let $M = \mathbb{R}^3 \setminus B\_1$ where $B\_1$ is unit ball. I am trying to solve the following PDE for $f$:


$$\Delta f -\frac{ f }{r^2}+ \frac{ \left. f \right|\_{\partial M}}{r^2} = 0, \qquad \text{on} \, M$$
$$f + a \partial\_r f = h, \qquad \text{on} \, \partial M$$
$$\lim\_{|x|\to \infty} f= 0 $$


where $a>0$ is a constant, $r = |x|$, $h \in L^2(\partial M)$ (or any nice function space you want). Also, $\left. f \right|\_{\partial M}$ is the function $f$ restricted to $r=1$; you can see it as $f(\frac{x}{|x|})$.


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\_{x\in \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?



Is this PDE solvable?
One can try the Kelvin transforn and expansion of solutions into series of spherical harmonics in $B\_1$. Since the equation is rotationally inveriant it should lead to simple enough equations for the harmonics' coefficients.


