Positive Eigenfunctions of a Schrödinger Operator

C. A. Stuart, Huan-Song Zhou

Type: Article

Publication Date: 2005-10-01

Citations: 8

DOI: https://doi.org/10.1112/s0024610705006873

Abstract

The paper considers the eigenvalue problem − Δ u − α u + λ g ( x ) u = 0 with u ∈ H 1 ( R N ) , u ≠ 0 , where ∞, λ ∈ and g x ≡ 0 on Ω ¯ , g ( x ) ∈ ( 0 , 1 ] on R N ∖ 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.

Locations

Journal of the London Mathematical Society - View
Infoscience (Ecole Polytechnique Fédérale de Lausanne) - View - PDF

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