We apply the compactness results obtained in the first part of this work, to prove existence and multiplicity results for finite energy solutions to the nonlinear elliptic equation
$$-\triangle u + V \left(\left|x\right|\right) u = g \left(\left|x\right|, u\right) \quad {\rm in} \Omega \subseteq \mathbb{R}^{N},\,N \geq 3,$$
where
\({\Omega}\) is a radial domain (bounded or unbounded) and
u satisfies
u = 0 on
\({\partial\Omega}\) if
\({\Omega \neq\mathbb{R}^{N}}\) and
\({u \rightarrow 0}\) as
\({\left|x\right| \rightarrow \infty}\) if
\({\Omega}\) is unbounded. The potential
V may be vanishing or unbounded at zero or at infinity and the nonlinearity
g may be superlinear or sublinear. If
g is sublinear, the case with a forcing term
\({g\left(\left|\cdot\right|, 0\right) \neq 0}\) is also considered. Our results allow to deal with
V and
g exhibiting behaviours at zero or at infinity which are new in the literature and, when
\({\Omega = \mathbb{R}^{N}}\), do not need to be compatible with each other.