Axisymmetric TE-modes in a self-focusing dielectric

We extend and improve earlier work on the existence of positive solutions u is an element of H-0(1) (0, infinity) of the nonlinear eigenvalue problem u ''(r) - 3/4r(2) u( r) + g (r, u(r)/root r) u(r) - k(2)u(r) = 0 for r > 0, where g is an element of C([0,infinity)(2)) is such that (i) 0 < A <= g( r, s) <= B < infinity for all r, s >= 0, and (ii) g( r, s) is a nondecreasing function of s for each r >= 0. This problem is central to the study of guided TE-modes propagating in an axisymmetric, self-focusing dielectric such as an optical fiber. The function g is a nonlinear refractive index and we are now able to dispense with unnecessary restrictions about its behavior.