In this talk, we study asymptotically and numerically the nonlinear eigenvalue problems under constraints arising from Bose-Einstein condensation, nonlinear optics, quantum physics and chemistry. We begin with the time-independent nonlinear Schrodinger equation also known as Gross-Pitaevskii equation and reformulated it into singular
perturbed nonlinear eigenvalue problems under constraints. Matched asymptotic approximations for the eigenfunctions and eigenvalues as well as the corresponding energy are presented in strongly interaction regimes. Boundary and/or interior layers and their width are presented. An efficient and accurate numerical method based on the normalized gradient flow is proposed for computing the first and/or other eigenfunctions of the nonlinear eigenvalue problems. Numerical results in one dimension (1D), 2D and 3D are reported. Finally, the analysis results and numerical method are extended to nonlinear eigenvalue problems of system of nonlinear Schrodinger equations.