  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 $\sigma\ge0$. 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 $\sigma\to\infty$ 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.
