coupled non linear different equations in matrix form

Question

Consider following differential equation of the form, $[\frac{d^2}{dx_1^2}+\frac{d^2}{dx_2^2}+x_1^2+x_2^2+gx_1^2x_2^2+x_1^4+x_2^4]u(x_1,x_2)=E u(x_1,x_2),$ where $g$ is the coupling constant. Please can anybody tell me what is an efficient algorithm to solve the system for $u(x_1,x_2)$. ($E$ are the Eigenvalues and $u(x_1,x_2)$ are the eigenfunctions).