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 - ▵ u + V x u = g x , u in Ω ⊆ R N , N ≥ 3 , where Ω is a radial domain (bounded or unbounded) and u satisfies u = 0 on ∂ Ω if Ω ≠ R N and u → 0 as x → ∞ if Ω 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 · , 0 ≠ 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 Ω = R N , do not need to be compatible with each other.