Semi-classical states for the Choquard equation
We study the nonlocal equation - ε 2 Δ u ε + V u ε = ε - α ( I α ∗ | u ε | p ) | u ε | p - 2 u ε in R N , where N ≥ 1 , α ∈ ( 0 , N ) , I α ( x ) = A α / | x | N - α is the Riesz potential and ε > 0 is a small parameter. We show that if the external potential V ∈ C ( R N ; [ 0 , ∞ ) ) has a local...
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| Veröffentlicht in: | Calculus of variations and partial differential equations Jg. 52; H. 1-2; S. 199 - 235 |
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| Format: | Journal Article |
| Sprache: | Englisch |
| Veröffentlicht: |
Berlin/Heidelberg
Springer Berlin Heidelberg
01.01.2015
Springer Nature B.V |
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| ISSN: | 0944-2669, 1432-0835 |
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| Abstract | We study the nonlocal equation
-
ε
2
Δ
u
ε
+
V
u
ε
=
ε
-
α
(
I
α
∗
|
u
ε
|
p
)
|
u
ε
|
p
-
2
u
ε
in
R
N
,
where
N
≥
1
,
α
∈
(
0
,
N
)
,
I
α
(
x
)
=
A
α
/
|
x
|
N
-
α
is the Riesz potential and
ε
>
0
is a small parameter. We show that if the external potential
V
∈
C
(
R
N
;
[
0
,
∞
)
)
has a local minimum and
p
∈
[
2
,
(
N
+
α
)
/
(
N
-
2
)
+
)
then for all small
ε
>
0
the problem has a family of solutions concentrating to the local minimum of
V
provided that: either
p
>
1
+
max
(
α
,
α
+
2
2
)
/
(
N
-
2
)
+
, or
p
>
2
and
lim inf
|
x
|
→
∞
V
(
x
)
|
x
|
2
>
0
, or
p
=
2
and
inf
x
∈
R
N
V
(
x
)
(
1
+
|
x
|
N
-
α
)
>
0
. Our assumptions on the decay of
V
and admissible range of
p
≥
2
are optimal. The proof uses variational methods and a novel nonlocal penalization technique that we develop in this work. |
|---|---|
| AbstractList | (ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image).We study the nonlocal equation ... ...where ..., ..., ... is the Riesz potential and ... is a small parameter. We show that if the external potential ... has a local minimum and ... then for all small ... the problem has a family of solutions concentrating to the local minimum of ... provided that: either ..., or ... and ..., or ... and ... Our assumptions on the decay of ... and admissible range of ... are optimal. The proof uses variational methods and a novel nonlocal penalization technique that we develop in this work. We study the nonlocal equation - ε 2 Δ u ε + V u ε = ε - α ( I α ∗ | u ε | p ) | u ε | p - 2 u ε in R N , where N ≥ 1 , α ∈ ( 0 , N ) , I α ( x ) = A α / | x | N - α is the Riesz potential and ε > 0 is a small parameter. We show that if the external potential V ∈ C ( R N ; [ 0 , ∞ ) ) has a local minimum and p ∈ [ 2 , ( N + α ) / ( N - 2 ) + ) then for all small ε > 0 the problem has a family of solutions concentrating to the local minimum of V provided that: either p > 1 + max ( α , α + 2 2 ) / ( N - 2 ) + , or p > 2 and lim inf | x | → ∞ V ( x ) | x | 2 > 0 , or p = 2 and inf x ∈ R N V ( x ) ( 1 + | x | N - α ) > 0 . Our assumptions on the decay of V and admissible range of p ≥ 2 are optimal. The proof uses variational methods and a novel nonlocal penalization technique that we develop in this work. |
| Author | Van Schaftingen, Jean Moroz, Vitaly |
| Author_xml | – sequence: 1 givenname: Vitaly surname: Moroz fullname: Moroz, Vitaly organization: Department of Mathematics, Swansea University – sequence: 2 givenname: Jean surname: Van Schaftingen fullname: Van Schaftingen, Jean email: Jean.VanSchaftingen@uclouvain.be organization: Institut de Recherche en Mathématique et Physique, Université Catholique de Louvain |
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| Snippet | We study the nonlocal equation
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-
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where
N
≥
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|... (ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image) We study the nonlocal equation ...where ..., ..., ... is the Riesz potential and... (ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image).We study the nonlocal equation ... ...where ..., ..., ... is the Riesz potential... |
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| SubjectTerms | Analysis Calculus of variations Calculus of Variations and Optimal Control; Optimization Control Decay Euclidean space Mathematical analysis Mathematical and Computational Physics Mathematics Mathematics and Statistics Optimization Partial differential equations Proving Schrodinger equation Systems Theory Texts Theoretical Variational methods |
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| Title | Semi-classical states for the Choquard equation |
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