  In this paper we look for standing waves for nonlinear Schr\"odinger
equations $$ i\frac{\partial \psi}{\partial t}+\Delta \psi - g(|y|) \psi
-W^{\prime}(| \psi |)\frac{\psi}{| \psi |}=0 $$ with cylindrically symmetric
potentials $g$ vanishing at infinity and non-increasing, and a $C^1$ nonlinear
term satisfying weak assumptions. In particular we show the existence of
standing waves with non-vanishing angular momentum with prescribed $L^2$ norm.
The solutions are obtained via a minimization argument, and the proof is given
for an abstract functional which presents lack of compactness. As a particular
case we prove the existence of standing waves with non-vanishing angular
momentum for the nonlinear hydrogen atom equation.
