  We provide a precise description of the set of normalized ground state
solutions (NGSS) for the class of elliptic equations:
  $$
  -\Delta u - \lambda u + V (| x |) u - f (| x |, u) = 0,\quad\text{in}\quad
\mathbb{R}^n,\ n\geq 1.
  $$
  In particular, we show that under suitable assumptions on $V$ and $f$, the
NGSS is unique for all the masses except for at most a finite number. Moreover,
we prove that when unique, the NGSS $u_c$ is a smooth function of the mass $c.$
Our method is as follow: using the NGSS for a given mass $c$, we construct an
exhaustive list of potential candidates to the minimization problem for masses
close to $c$, and we develop a strategy how to pick the right one. In
particular, if there is a unique NGSS for a given mass $c_0,$ then this
uniqueness property is inherited for all the masses $c$ close to $c_0.$ Our
method is general and applies to other equations provided that some key
properties hold true.
