Global, small radially symmetric solutions to nonlinear Schrdinger equations and a gauge transformation

This paper proves the global existence of small radially symmetric solu­tions to the nonlinear Schrodinger equations of the form { i8,u + ½.6.u = F(u, Vu, ii, Vu), (t, :z:) nE x n, u(O, :z:) = eo¢(1:z:I), :z: E , where n >= 3, eo is sufficiently small, l:z:I = ( E :z:])1 12 , L >-au°'lu°'2 lo$lal91 +L >.a13u°'lu°'2( L l8;ul2)131( L (8;u)2)132 with >.a,>..013 E C, l1,l2,l3 E N, lo = 3 for n = 3,4, and lo = 2 for n 5. The method depends on the combination of a gauge transformation and generalized energy estimtes and does not require the condition such that 8vu F is pure imaginary which is needed for the classical energy method