Ground state solution for a class of Kirchhoff-type equation with general convolution nonlinearity
In this paper, we consider the following class of Kirchhoff-type equation - ( a + b ∫ R N | ∇ u | 2 d x ) Δ u + V ( x ) u = ( I α ∗ F ( u ) ) f ( u ) , in R N , u ∈ H 1 ( R N ) , where a > 0 , b ≥ 0 , N ≥ 3 , α ∈ ( N - 2 , N ) , V : R N → R is a potential function and I α is a Riesz potential of...
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| Published in | Zeitschrift für angewandte Mathematik und Physik Vol. 73; no. 2 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Cham
Springer International Publishing
01.04.2022
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
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| Abstract | In this paper, we consider the following class of Kirchhoff-type equation
-
(
a
+
b
∫
R
N
|
∇
u
|
2
d
x
)
Δ
u
+
V
(
x
)
u
=
(
I
α
∗
F
(
u
)
)
f
(
u
)
,
in
R
N
,
u
∈
H
1
(
R
N
)
,
where
a
>
0
,
b
≥
0
,
N
≥
3
,
α
∈
(
N
-
2
,
N
)
,
V
:
R
N
→
R
is a potential function and
I
α
is a Riesz potential of order
α
∈
(
N
-
2
,
N
)
. Under certain assumptions on
V
(
x
) and
f
(
u
), we prove that the equation has ground state solutions by variational methods. |
|---|---|
| AbstractList | In this paper, we consider the following class of Kirchhoff-type equation -(a+b∫RN|∇u|2dx)Δu+V(x)u=(Iα∗F(u))f(u),inRN,u∈H1(RN),where a>0, b≥0, N≥3, α∈(N-2,N), V:RN→R is a potential function and Iα is a Riesz potential of order α∈(N-2,N). Under certain assumptions on V(x) and f(u), we prove that the equation has ground state solutions by variational methods. In this paper, we consider the following class of Kirchhoff-type equation - ( a + b ∫ R N | ∇ u | 2 d x ) Δ u + V ( x ) u = ( I α ∗ F ( u ) ) f ( u ) , in R N , u ∈ H 1 ( R N ) , where a > 0 , b ≥ 0 , N ≥ 3 , α ∈ ( N - 2 , N ) , V : R N → R is a potential function and I α is a Riesz potential of order α ∈ ( N - 2 , N ) . Under certain assumptions on V ( x ) and f ( u ), we prove that the equation has ground state solutions by variational methods. |
| ArticleNumber | 75 |
| Author | Zhou, Li Zhu, Chuanxi |
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| References_xml | – volume: 2 start-page: 787 issue: 129 year: 1999 end-page: 809 ident: CR2 article-title: On the existence of bounded Palais–Snale sequences and application to a Landsman–Lazer-type problem set on publication-title: Proc. Edinb. Math. Soc. doi: 10.1017/S0308210500013147 contributor: fullname: Jeanjean – year: 2001 ident: CR12 publication-title: Analysis contributor: fullname: Loss – year: 1996 ident: CR14 publication-title: Minimax Theorems, Progress in Nonlinear Differential Equations and Their Applications contributor: fullname: Willem – volume: 254 start-page: 3089 year: 2013 end-page: 3145 ident: CR6 article-title: Nonexistence and optimal decay of supersolutions to Choquard equations in exterior domains publication-title: J. Differ. Equ. doi: 10.1016/j.jde.2012.12.019 contributor: fullname: Van Schaftingen – volume: 195 start-page: 455 year: 2010 end-page: 467 ident: CR10 article-title: Classification of positive solitary solutions of the nonlinear Choquard equation publication-title: Arch Ration. Mech. Aral doi: 10.1007/s00205-008-0208-3 contributor: fullname: Lin – ident: CR4 – volume: 467 start-page: 842 year: 2018 end-page: 862 ident: CR11 article-title: Ground state solutions of Poho aev type and Nehari type for a class of nonlinear Choquard equations publication-title: J. Math. Anal. Appl. doi: 10.1016/j.jmaa.2018.07.055 contributor: fullname: Luo – volume: 17 start-page: 1550005 year: 2015 ident: CR7 article-title: Ground states of nonlinear Choquard equations: Hardy–Littlewood–Sobolev critical exponent publication-title: Commun. Contemp. Math. doi: 10.1142/S0219199715500054 contributor: fullname: Van Schaftingen – volume: 271 start-page: 107 year: 2016 end-page: 135 ident: CR9 article-title: Nodal solutions for the Choquard equation publication-title: J. Funct. Anal. doi: 10.1016/j.jfa.2016.04.019 contributor: fullname: Van Schaftingen – ident: CR13 – volume: 259 start-page: 2884 year: 2015 end-page: 2902 ident: CR1 article-title: Ground states for Kirchhoff equations without compact condition publication-title: J. Differ. Equ. doi: 10.1016/j.jde.2015.04.005 contributor: fullname: Guo – volume: 15 start-page: 2733 year: 1998 end-page: 2742 ident: CR3 article-title: Spherically-symmetric solutions of Schrödinger–Newton equations publication-title: Class. Quantum Gravity doi: 10.1088/0264-9381/15/9/019 contributor: fullname: Tod – ident: CR5 – volume: 19 start-page: 773 year: 2017 end-page: 813 ident: CR8 article-title: A guide to the Choquard equation publication-title: J. Fixed Point Theory Appl. doi: 10.1007/s11784-016-0373-1 contributor: fullname: Van Schaftingen – ident: 1712_CR4 doi: 10.1016/j.jfa.2013.04.007 – volume: 259 start-page: 2884 year: 2015 ident: 1712_CR1 publication-title: J. Differ. Equ. doi: 10.1016/j.jde.2015.04.005 contributor: fullname: Z Guo – volume: 2 start-page: 787 issue: 129 year: 1999 ident: 1712_CR2 publication-title: Proc. Edinb. Math. Soc. doi: 10.1017/S0308210500013147 contributor: fullname: L Jeanjean – volume: 467 start-page: 842 year: 2018 ident: 1712_CR11 publication-title: J. Math. Anal. Appl. doi: 10.1016/j.jmaa.2018.07.055 contributor: fullname: HX Luo – volume-title: Analysis year: 2001 ident: 1712_CR12 doi: 10.1090/gsm/014 contributor: fullname: EH Lieb – ident: 1712_CR5 doi: 10.1090/S0002-9947-2014-06289-2 – volume-title: Minimax Theorems, Progress in Nonlinear Differential Equations and Their Applications year: 1996 ident: 1712_CR14 contributor: fullname: M Willem – volume: 15 start-page: 2733 year: 1998 ident: 1712_CR3 publication-title: Class. Quantum Gravity doi: 10.1088/0264-9381/15/9/019 contributor: fullname: IM Moroz – volume: 195 start-page: 455 year: 2010 ident: 1712_CR10 publication-title: Arch Ration. Mech. Aral doi: 10.1007/s00205-008-0208-3 contributor: fullname: L Ma – volume: 17 start-page: 1550005 year: 2015 ident: 1712_CR7 publication-title: Commun. Contemp. Math. doi: 10.1142/S0219199715500054 contributor: fullname: V Moroz – volume: 19 start-page: 773 year: 2017 ident: 1712_CR8 publication-title: J. Fixed Point Theory Appl. doi: 10.1007/s11784-016-0373-1 contributor: fullname: V Moroz – ident: 1712_CR13 doi: 10.1007/BF00250555 – volume: 271 start-page: 107 year: 2016 ident: 1712_CR9 publication-title: J. Funct. Anal. doi: 10.1016/j.jfa.2016.04.019 contributor: fullname: M Chimenti – volume: 254 start-page: 3089 year: 2013 ident: 1712_CR6 publication-title: J. Differ. Equ. doi: 10.1016/j.jde.2012.12.019 contributor: fullname: V Moroz |
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| Snippet | In this paper, we consider the following class of Kirchhoff-type equation
-
(
a
+
b
∫
R
N
|
∇
u
|
2
d
x
)
Δ
u
+
V
(
x
)
u
=
(
I
α
∗
F
(
u
)
)
f
(
u
)
,
in
R
N... In this paper, we consider the following class of Kirchhoff-type equation -(a+b∫RN|∇u|2dx)Δu+V(x)u=(Iα∗F(u))f(u),inRN,u∈H1(RN),where a>0, b≥0, N≥3, α∈(N-2,N),... |
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| SubjectTerms | Engineering Ground state Mathematical Methods in Physics Theoretical and Applied Mechanics Variational methods |
| Title | Ground state solution for a class of Kirchhoff-type equation with general convolution nonlinearity |
| URI | https://link.springer.com/article/10.1007/s00033-022-01712-0 https://www.proquest.com/docview/2640560199/abstract/ |
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