Nonlinear Schrödinger equations near an infinite well potential

The paper deals with standing wave solutions of the dimensionless nonlinear Schrödinger equation where the potential V λ : R N → R is close to an infinite well potential V ∞ : R N → R , i. e. V ∞ = ∞ on an exterior domain R N \ Ω , V ∞ | Ω ∈ L ∞ ( Ω ) , and V λ → V ∞ as λ → ∞ in a sense to be made p...

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Published inCalculus of variations and partial differential equations Vol. 51; no. 1-2; pp. 363 - 379
Main Authors Bartsch, Thomas, Parnet, Mona
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer Berlin Heidelberg 01.09.2014
Springer Nature B.V
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ISSN0944-2669
1432-0835
DOI10.1007/s00526-013-0678-5

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Summary:The paper deals with standing wave solutions of the dimensionless nonlinear Schrödinger equation where the potential V λ : R N → R is close to an infinite well potential V ∞ : R N → R , i. e. V ∞ = ∞ on an exterior domain R N \ Ω , V ∞ | Ω ∈ L ∞ ( Ω ) , and V λ → V ∞ as λ → ∞ in a sense to be made precise. The nonlinearity may be of Gross–Pitaevskii type. A standing wave solution of ( N L S λ ) with λ = ∞ vanishes on R N \ Ω and satisfies Dirichlet boundary conditions, hence it solves We investigate when a standing wave solution Φ ∞ of the infinite well potential ( N L S ∞ ) gives rise to nearby solutions Φ λ of the finite well potential ( N L S λ ) with λ ≫ 1 large. Considering ( N L S ∞ ) as a singular limit of ( N L S λ ) we prove a kind of singular continuation type results.
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ISSN:0944-2669
1432-0835
DOI:10.1007/s00526-013-0678-5