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 |
|---|---|
| Hauptverfasser: | , |
| Format: | Journal Article |
| Sprache: | Englisch |
| Veröffentlicht: |
Berlin/Heidelberg
Springer Berlin Heidelberg
01.01.2015
Springer Nature B.V |
| Schlagworte: | |
| ISSN: | 0944-2669, 1432-0835 |
| Online-Zugang: | Volltext |
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| Zusammenfassung: | 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. |
|---|---|
| Bibliographie: | SourceType-Scholarly Journals-1 ObjectType-Feature-1 content type line 14 ObjectType-Article-1 ObjectType-Feature-2 content type line 23 |
| ISSN: | 0944-2669 1432-0835 |
| DOI: | 10.1007/s00526-014-0709-x |