Nonlinear Schrِdinger equations with multiple-well potential

We consider the stationary solutions for a class of Schrِdinger equations with a N -well potential and a nonlinear perturbation. By means of semiclassical techniques we prove that the dominant term of the ground state solutions is described by a N -dimensional Hamiltonian system, where the coupling term among the coordinates is a tridiagonal Toeplitz matrix. In particular, in the limit of large focusing nonlinearity we prove that the ground state stationary solutions consist of N wavefunctions localized on a single well. Furthermore, we consider in detail the case of N = 4 wells, where we show the occurrence of spontaneous symmetry-breaking bifurcation effect.