I'm struggling with the following elliptic PDE:
$$ \left(\frac{\partial^2}{\partial r_a^2} + \frac{\partial^2}{\partial r_b^2}\right) f(r_a, r_b) + 2\left( \frac{1}{r_a} -\frac{1}{a_0} \right) \frac{\partial}{\partial r_a} f(r_a, r_b) + 2\left( \frac{1}{r_b} -\frac{1}{a_0} \right) \frac{\partial}{\partial r_b} f(r_a, r_b) - \left(\frac{2}{r_a ^2} +\frac{2}{r_b ^2}\right) f(r_a, r_b) + \frac{4 r_a r_b}{R^3} =0 $$
Here, $a_0$ and $R$ are fixed parameters. I have also tried to use relative coordinates $$ r_1 = \frac{r_a + r_b}{2} $$ and $$ r_2 = \frac{r_a + r_b}{2} $$ in order to decouple the mixed term (namely $4 r_a r_b/R^3$) but then another coupling between $r_a$ and $r_b$ appears again within the first order derivative coefficients and in the term that is multiplying $f(r_a, r_b)$. Any hint?
NB: I was able to numerically integrate the equation but providing an analyitical solution would be amazing.