Compactness and existence results in weighted Sobolev spaces of radial functions. Part II: existence

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 -Delta u + V(vertical bar x vertical bar) u = g(vertical bar x vertical bar) u = g (vertical bar x vertical bar, u) in Omega subset of R-N, N >= 3, where Omega is a radial domain (bounded or unbounded) and u satisfies u = 0 on partial derivative Omega if Omega not equal R-N and u -> 0 as vertical bar x vertical bar -> infinity 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 (vertical bar.vertical bar, 0) not equal 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 = R-N, do not need to be compatible with each other.