I am attempting to solve a PDE, namely the Eikonal equation in two dimensions, perturbatively, because the scenario I am solving presents itself in a perturbative manner. The form of the PDE is $$\left(\frac{\partial u}{\partial r}\right)^2 +\frac{1}{r^2} \left(\frac{\partial u}{\partial \theta}\right)^2= f(r,\theta) + \epsilon g(r,\theta)$$
where both $f(r,\theta)$ and $g(r,\theta)$ are singular for some values of $(r,\theta)$ (which are not the same for $f$ and $g$). I know the solution for
$$\left(\frac{\partial u}{\partial r}\right)^2 +\frac{1}{r^2} \left(\frac{\partial u}{\partial \theta}\right)^2= f(r,\theta)$$
and have introduced the function $g(r,\theta)$ as a perturbation with the small parameter $\epsilon$. When I naively applied perturbation theory to it, I got a lot of non-sensical aspects to the solution. I recently read about singular perturbation theory and I was wondering if that is what would be applicable here. Any resource that I came across doesn't feature PDEs in such form being solved with singular perturbation theory.
If anyone can tell me that it is actually the singular perturbation theory that I need to apply, then it would be great. As much as I could read about it, I couldn't solve my problem with that. Can anyone suggest a solution or point towards a method or resource that solves a similar problem?
Also, any good resource on the same would be helpful. Thanks!
P.S. $g(r,\theta)$ looks like $\frac{1}{(r^2+d^2-2rd\cos{\theta})^2}$ where $d$ is a constant.