Consider $\Big( \big(\nabla^2 + \alpha\big)^2 + \beta\Big) f(r,\theta,\phi) = 0$, where $\alpha$ and $\beta$ are positive constants. The boundary condition is $\frac{\partial^n f}{\partial r^n} = 0$ for $n=0,...,3$ as $r \to \infty $.
I tried to use separation of variables to solve this, which is complicated as the $\nabla^4$ term makes it challenging to separate the variables. I wonder if there is an alternative method to solve this equation. Thank you in advance for your help.