Normalized solutions of mass supercritical Schrödinger–Poisson equation with potential
In this paper we prove the existence of normalized solutions ( λ , u ) ⊂ ( 0 , ∞ ) × H 1 ( R 3 ) to the following Schrödinger–Poisson equation - Δ u + V ( x ) u + λ u + ( | x | - 1 ∗ u 2 ) u = | u | p - 2 u in R 3 , u > 0 , ∫ R 3 u 2 d x = a 2 , where a > 0 is fixed, p ∈ ( 10 3 , 6 ) is a give...
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| Published in | Calculus of variations and partial differential equations Vol. 64; no. 5 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Berlin/Heidelberg
Springer Berlin Heidelberg
01.06.2025
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
| ISSN | 0944-2669 1432-0835 |
| DOI | 10.1007/s00526-025-03012-7 |
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| Abstract | In this paper we prove the existence of normalized solutions
(
λ
,
u
)
⊂
(
0
,
∞
)
×
H
1
(
R
3
)
to the following Schrödinger–Poisson equation
-
Δ
u
+
V
(
x
)
u
+
λ
u
+
(
|
x
|
-
1
∗
u
2
)
u
=
|
u
|
p
-
2
u
in
R
3
,
u
>
0
,
∫
R
3
u
2
d
x
=
a
2
,
where
a
>
0
is fixed,
p
∈
(
10
3
,
6
)
is a given exponent and the potential
V
satisfies some suitable conditions. Since the
L
2
(
R
3
)
-norm of
u
is fixed,
λ
appears as a Lagrange multiplier. For
V
(
x
)
≥
0
, our solutions are obtained by using a mountain-pass argument on bounded domains and a limit process introduced by Bartsch et al (Commun Partial Differ Equ 46:1729–1756, 2021). For
V
(
x
)
≤
0
, we directly construct an entire mountain-pass solution with positive energy. |
|---|---|
| AbstractList | In this paper we prove the existence of normalized solutions (λ,u)⊂(0,∞)×H1(R3) to the following Schrödinger–Poisson equation -Δu+V(x)u+λu+(|x|-1∗u2)u=|u|p-2uinR3,u>0,∫R3u2dx=a2,where a>0 is fixed, p∈(103,6) is a given exponent and the potential V satisfies some suitable conditions. Since the L2(R3)-norm of u is fixed, λ appears as a Lagrange multiplier. For V(x)≥0, our solutions are obtained by using a mountain-pass argument on bounded domains and a limit process introduced by Bartsch et al (Commun Partial Differ Equ 46:1729–1756, 2021). For V(x)≤0, we directly construct an entire mountain-pass solution with positive energy. In this paper we prove the existence of normalized solutions ( λ , u ) ⊂ ( 0 , ∞ ) × H 1 ( R 3 ) to the following Schrödinger–Poisson equation - Δ u + V ( x ) u + λ u + ( | x | - 1 ∗ u 2 ) u = | u | p - 2 u in R 3 , u > 0 , ∫ R 3 u 2 d x = a 2 , where a > 0 is fixed, p ∈ ( 10 3 , 6 ) is a given exponent and the potential V satisfies some suitable conditions. Since the L 2 ( R 3 ) -norm of u is fixed, λ appears as a Lagrange multiplier. For V ( x ) ≥ 0 , our solutions are obtained by using a mountain-pass argument on bounded domains and a limit process introduced by Bartsch et al (Commun Partial Differ Equ 46:1729–1756, 2021). For V ( x ) ≤ 0 , we directly construct an entire mountain-pass solution with positive energy. |
| ArticleNumber | 152 |
| Author | Rizzi, Matteo Peng, Xueqin |
| Author_xml | – sequence: 1 givenname: Xueqin orcidid: 0000-0002-6595-7693 surname: Peng fullname: Peng, Xueqin email: pxq52918@163.com organization: Department of Mathematical Sciences, Tsinghua University – sequence: 2 givenname: Matteo surname: Rizzi fullname: Rizzi, Matteo organization: Mathematisches Institut, Justus-Liebig-University Giessen |
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| Cites_doi | 10.1112/plms/pds072 10.1007/s00033-010-0092-1 10.1007/BF00282048 10.1017/CBO9780511551703 10.1016/j.jmaa.2014.02.038 10.1016/j.jfa.2006.04.005 10.1016/S0362-546X(96)00021-1 10.1007/s00033-012-0272-2 10.4310/jdg/1214433725 10.1007/s00526-002-0169-6 10.1007/BF00250556 10.1090/S0002-9939-1983-0699419-3 10.1016/j.anihpc.2004.07.005 10.1007/s00033-021-01633-4 10.1007/s00526-014-0724-y 10.1016/j.jmaa.2008.04.053 10.1080/03605302.2021.1893747 10.1007/BF01205672 10.1016/j.jfa.2011.06.014 10.1007/s00013-012-0468-x 10.1090/cbms/065 10.1007/BF02418013 10.1016/j.jmaa.2019.123447 10.1016/j.jmaa.2020.124071 10.1016/j.jde.2021.09.022 10.1016/0001-8708(77)90108-6 10.1016/S0893-9659(01)80038-0 10.1007/s00208-024-02857-1 10.1088/1361-6544/ad1efb 10.1007/s00033-022-01922-6 10.1016/j.jde.2022.06.012 |
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| References | H Brézis (3012_CR14) 1983; 88 V Benci (3012_CR10) 1976; 13 L Nirenberg (3012_CR31) 1959; 13 J Bellazzini (3012_CR8) 2011; 62 L Zhao (3012_CR39) 2008; 346 V Benci (3012_CR9) 1987; 99 H Berestycki (3012_CR11) 1983; 82 T Bartsch (3012_CR4) 2024; 390 TJ Luo (3012_CR28) 2014; 416 J Liu (3012_CR27) 2015; 52 J Bellazzini (3012_CR6) 2013; 107 EH Lieb (3012_CR23) 2001 J Jeanjean (3012_CR19) 1997; 28 R Molle (3012_CR30) 2022; 333 XQ Peng (3012_CR33) 2023; 74 T Bartsch (3012_CR2) 2013; 100 L Jeanjean (3012_CR21) 2013; 64 PL Lions (3012_CR25) 1987; 109 3012_CR34 T Aubin (3012_CR1) 1976; 11 3012_CR13 G Talenti (3012_CR37) 1976; 110 3012_CR38 3012_CR17 T Bartsch (3012_CR5) 2005; 22 3012_CR18 N Li (3012_CR22) 2020; 488 NJ Mauser (3012_CR29) 2001; 14 XQ Peng (3012_CR32) 2021; 72 T Bartsch (3012_CR3) 2021; 46 P Pucci (3012_CR26) 2024; 37 J Borthwick (3012_CR12) 2024; 377 M Struwe (3012_CR36) 1996 ST Chen (3012_CR16) 2020; 481 D Ruiz (3012_CR35) 2006; 237 EH Lieb (3012_CR24) 1977; 23 G Cerami (3012_CR15) 2003; 17 J Bellazzini (3012_CR7) 2011; 261 L Jeanjean (3012_CR20) 2021; 303 |
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| Snippet | In this paper we prove the existence of normalized solutions
(
λ
,
u
)
⊂
(
0
,
∞
)
×
H
1
(
R
3
)
to the following Schrödinger–Poisson equation
-
Δ
u
+
V
(
x
)... In this paper we prove the existence of normalized solutions (λ,u)⊂(0,∞)×H1(R3) to the following Schrödinger–Poisson equation... |
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| SubjectTerms | Analysis Calculus of Variations and Optimal Control; Optimization Control Lagrange multiplier Mathematical and Computational Physics Mathematics Mathematics and Statistics Poisson equation Systems Theory Theoretical |
| Title | Normalized solutions of mass supercritical Schrödinger–Poisson equation with potential |
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