Ground states and energy asymptotics of the nonlinear Schr\"{o}dinger equation

We study analytically the existence and uniqueness of the ground state of the nonlinear Schr"{o}dinger equation (NLSE) with a general power nonlinearity described by the power index $\sigma \ge 0$. For the NLSE under a box or a harmonic potential, we can derive explicitly the approximations of the ground states and their corresponding energy and chemical potential in weak or strong interaction regimes with a fixed nonlinearity $\sigma$. Besides, we study the case where the nonlinearity $\sigma \to \mathrm{\infty }$ with a fixed interaction strength. In particular, a bifurcation in the ground states is observed. Numerical results in 1D and 2D will be reported to support our asymptotic results.