Solution to the "cubic" Helmholtz equation What is known about the solutions of the differential equation in three-dimensions
$$
\nabla^2 \phi = -\kappa^2 (\phi  + (1/3!)\phi^3)
$$
Without the cubic term, this gives a linear operator $\mathcal{L} = \nabla^2 + \kappa^2$. In this case I can get a solution via the Green's function $G=\exp{(i\kappa r)}(4\pi r)^{-1}$. In my equation however, the presence of $\phi^3$ does not give me a linear operator. Is anything known about the solution to this equation?
Context: The Poisson-Boltzmann equation can be put into the functional form of $\nabla^2 \phi = -\kappa^2 \sinh \phi$. Expanding sinh to first order gives the Helmholtz equation as mentioned above. The second order term is zero and the third order term gives the equation in question.

This paper discusses your problem below Eq. 8 and provides the solution in Eq. 9 and 10 for a single plate and Eq. 11 and 12 for two plates.
