Abstract :
The paper considers the eigenvalue problem
−Δu − αu + λg(x)u = 0 with u ∈ H1(RN ), u = 0,
where α, λ ∈ R and
g(x) ≡ 0 on Ω, g(x) ∈ (0, 1] on RN \ 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 discussed.
