CONSTRAINT MINIMIZERS OF PERTURBED GROSS-PITAEVSKII ENERGY FUNCTIONALS IN RN

This paper is concerned with constraint minimizers of an L-2 - critical minimization problem (1) in R-N (N >= 1) under an L-2 - subcritical perturbation. We prove that the problem admits minimizers with mass rho(N/2) if and only if 0 <= rho < rho* := parallel to Q parallel to(4/N)(2) for b >= 0 and 0 < rho <= rho* for b < 0, where the constant b comes from the coefficient of the perturbation term, and Q is the unique positive radically symmetric solution of Delta u(x) - u(x) + u(1+4N)(x) = 0 in R-N. Furthermore, we analyze rigorously the concentration behavior of minimizers as rho NE arrow rho* for the case where b > 0, which shows that the concentration rates are determined by the subcritical perturbation, instead of the local profiles of the potential V(x).