On the second eigenvalue of a class of Schrödinger operators

In this paper, for each λ>0λ>0, we discuss the existence and some properties of the second eigenvalue α2(λ)α2(λ) for the following eigenvalue problem −Δu+λg(x)u=αu,u∈H1(RN),N≥3, where λ,α∈Rλ,α∈R and g(x)≡0on Ω¯,g(x)∈(0,1]on RN∖Ω¯andlim|x|→∞g(x)=1 for some bounded open set Ω⊂RNΩ⊂RN.