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) inΩ⊆RN N≥3, where Ω is a radial domain (bounded or unbounded) and u satisfies u = 0 on ∂Ω if Ω ≠ RN 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 = RN, do not need to be compatible with each other.