The Galerkin method can definitely be used for nonlinear problems.
I have seen often that one can couple Galerkin method with a fixed point method. For example you have a nonlinear PDE which you can linearise. The linearised PDE you can solve maybe with a Galerkin method, and then one shows that there is a fixed point of the appropriate map that takes the part you made linear to the solution.
Here is a paper where a Galerkin method is applied to a porous medium equation.
Also people use a famous theorem by Crandall and Liggett for nonlinear PDEs where basically the PDE is discretised in time and one gets a series of elliptic problems. One then shows that the solutions to these problems converge in some sense to a mild solution if I recall correctly. But I think this has something to do with semigroups.