The paper considers the eigenvalue problem -Δu-αu+λg(x)u=0 withu∈H1(RN),u≠0 where ∞, λ ∈ and g(x)≡0 on Ω¯, g(x) ∈ (0,1] onRN \ Ω¯ and lim|x|→+∞g(x)=1 for some bounded open set Ω∈RN. Given α>0, does there exist a value of λ>0 for which the problem has a positive solution? It is shown that this occurs if and only if α lies in a certain interval (Γ,ξ1) and that in this case the value of λ is unique, λ=Λ(α). The properties of the function Λ(α) are also discusse
