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 in | Calculus of variations and partial differential equations Vol. 51; no. 1-2; pp. 363 - 379 |
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Main Authors | , |
Format | Journal Article |
Language | English |
Published |
Berlin/Heidelberg
Springer Berlin Heidelberg
01.09.2014
Springer Nature B.V |
Subjects | |
Online Access | Get full text |
ISSN | 0944-2669 1432-0835 |
DOI | 10.1007/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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Bibliography: | SourceType-Scholarly Journals-1 ObjectType-Feature-1 content type line 14 ObjectType-Article-2 content type line 23 |
ISSN: | 0944-2669 1432-0835 |
DOI: | 10.1007/s00526-013-0678-5 |