Nonlinear Schrödinger equation with power-law confining potential

Critical limits of a stationary nonlinear three-dimensional Schrödinger equation with confining power-law potentials (∼ rα) are obtained using spherical symmetry. When the nonlinearity is given by an attractive two-body interaction (negative cubic term), it is shown how the maximum number of particles Nc in the trap increases as α decreases. With a negative cubic and positive quintic terms we study a first order phase transition, that occurs if the strength g3 of the quintic term is less than a critical value g3c. At the phase transition, the behavior of g3c with respect to α is given by g3c ∼0.0036+0.0251α+0.0088/ α2. © 2006 The American Physical Society.