Using perturbation techniques to approximate a solution to a non-linear PDE

Question

I am currently trying to solve the non-linear PDE $$ \theta G = \frac{(q \partial_w G)^{-\overline{\delta}}}{\overline{\delta}} + \frac{(\partial_w G)^{-\overline{\epsilon}}}{\overline{\epsilon}} + (\rho w + J) \partial_w G + \beta q \partial_q G $$ where $\overline{\delta} = \delta / (1 - \delta)$ and similarly for $\overline{\epsilon}$. The greek letters are all constants. After some trial and error (and a lot of experimentation with Mathematica), I have found a solution in the case $\delta = \epsilon$, call it $G_0(q, w)$. I would now like to understand if it is possible to use perturbation techniques to approximate the solution in the general case. Concretely, I am looking for a way to write down something that might look like $$ G(q, w) = G_0(q, w) + g_1 (\delta-\epsilon) + g_2 (\delta-\epsilon)^2 + \cdots\ , $$ where the $g_i$'s are constants or functions to be determined. Any ideas or references in this direction would be much appreciated.