The objective of this work is to solve nonlinear Poisson problems, particularly in 3D, using an innovative combination of the Method of Fundamental Solutions (MFS) and the Asymptotic Numerical Method (ANM). Using perturbation methods, non-linear problems are transformed into a series of linear differential equations with variable coefficients. To compensate for the absence of fundamental solutions for certain variable-coefficient operators, the MFS method is combined with the Radial Basis Functions (RBF) and the Analogous Equations Method (AEM). This approach makes it possible to efficiently solve the resulting linear equations and to follow the branches of non-linear solutions, even in complex domains such as three-dimensional cubic and spherical geometries.