You can find the Navier' s solution for a non-linear problem. Because, the assumed bases (whatever you are choosing for field variable i.e. double fourier series) will not satisfy the governing equations of non-linear problem. Yes, you try Rayleigh Ritz or Galerkin type exact solution for non-linear problem......
you can use the two-step perturbation method to solve nonlinear equations, "A two-step perturbation method in nonlinear analysis of beams, plates and shell", (Hui-Shen Shen)