p-Laplacian Equations on Locally Finite Graphs
This paper is mainly concerned with the following nonlinear p -Laplacian equation − Δ p ( x ) + ( λ a ( x ) + 1 ) ∣ u ∣ p − 2 ( x ) u ( x ) = f ( x , u ( x ) ) , i n V on a locally finite graph G = ( V, E ) with more general nonlinear term, where Δ p is the discrete p -Laplacian on graphs, p ≥ 2. Un...
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| Published in | Acta mathematica Sinica. English series Vol. 37; no. 11; pp. 1645 - 1678 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Beijing
Institute of Mathematics, Chinese Academy of Sciences and Chinese Mathematical Society
01.11.2021
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
| ISSN | 1439-8516 1439-7617 |
| DOI | 10.1007/s10114-021-9523-5 |
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| Abstract | This paper is mainly concerned with the following nonlinear
p
-Laplacian equation
−
Δ
p
(
x
)
+
(
λ
a
(
x
)
+
1
)
∣
u
∣
p
−
2
(
x
)
u
(
x
)
=
f
(
x
,
u
(
x
)
)
,
i
n
V
on a locally finite graph
G
= (
V, E
) with more general nonlinear term, where Δ
p
is the discrete
p
-Laplacian on graphs,
p
≥ 2. Under some suitable conditions on
f
and
a
(
x
), we can prove that the equation admits a positive solution by the Mountain Pass theorem and a ground state solution
u
λ
via the method of Nehari manifold, for any λ > 1. In addition, as λ → +∞, we prove that the solution
u
λ
converge to a solution of the following Dirichlet problem
{
−
Δ
p
(
x
)
+
∣
u
∣
p
−
2
(
x
)
u
(
x
)
=
f
(
x
,
u
(
x
)
)
,
i
n
Ω
u
(
x
)
=
0
o
n
∂
Ω
where Ω = {
x
∈
V
:
a
(
x
) = 0} is the potential well and
∂
Ω denotes the the boundary of Ω. |
|---|---|
| AbstractList | This paper is mainly concerned with the following nonlinear p-Laplacian equation −Δp(x)+(λa(x)+1)∣u∣p−2(x)u(x)=f(x,u(x)),inV on a locally finite graph G = (V, E) with more general nonlinear term, where Δp is the discrete p-Laplacian on graphs, p ≥ 2. Under some suitable conditions on f and a(x), we can prove that the equation admits a positive solution by the Mountain Pass theorem and a ground state solution uλ via the method of Nehari manifold, for any λ > 1. In addition, as λ → +∞, we prove that the solution uλ converge to a solution of the following Dirichlet problem {−Δp(x)+∣u∣p−2(x)u(x)=f(x,u(x)),inΩu(x)=0on∂Ω where Ω = {x ∈ V:a(x) = 0} is the potential well and ∂Ω denotes the the boundary of Ω. This paper is mainly concerned with the following nonlinear p -Laplacian equation − Δ p ( x ) + ( λ a ( x ) + 1 ) ∣ u ∣ p − 2 ( x ) u ( x ) = f ( x , u ( x ) ) , i n V on a locally finite graph G = ( V, E ) with more general nonlinear term, where Δ p is the discrete p -Laplacian on graphs, p ≥ 2. Under some suitable conditions on f and a ( x ), we can prove that the equation admits a positive solution by the Mountain Pass theorem and a ground state solution u λ via the method of Nehari manifold, for any λ > 1. In addition, as λ → +∞, we prove that the solution u λ converge to a solution of the following Dirichlet problem { − Δ p ( x ) + ∣ u ∣ p − 2 ( x ) u ( x ) = f ( x , u ( x ) ) , i n Ω u ( x ) = 0 o n ∂ Ω where Ω = { x ∈ V : a ( x ) = 0} is the potential well and ∂ Ω denotes the the boundary of Ω. |
| Author | Shao, Meng Qiu Han, Xiao Li |
| Author_xml | – sequence: 1 givenname: Xiao Li surname: Han fullname: Han, Xiao Li organization: Department of Mathematical Sciences, Tsinghua University – sequence: 2 givenname: Meng Qiu surname: Shao fullname: Shao, Meng Qiu email: shaomq17@mails.tsinghua.edu.cn organization: Department of Mathematical Sciences, Tsinghua University |
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| Cites_doi | 10.1007/BF00946631 10.1016/j.aml.2018.04.005 10.1080/03605309208820848 10.1007/s11425-016-0422-y 10.1090/S0002-9947-1960-0111898-8 10.1016/j.jmaa.2017.07.028 10.1080/00036811.2013.842640 10.1090/proc/13929 10.1016/j.jde.2019.10.007 10.1016/j.jmaa.2012.04.026 10.1007/s00033-004-1084-9 10.1007/s00526-017-1204-y 10.1016/j.jde.2012.12.005 10.1016/j.na.2005.05.056 10.1016/j.jmaa.2015.10.024 10.1016/j.jmaa.2011.03.006 10.1007/PL00001511 10.1007/s11425-017-9254-7 10.1016/j.na.2015.11.014 10.1007/s00209-019-02253-5 10.1007/s00229-003-0397-x 10.1016/j.jmaa.2017.12.020 10.1063/1.3683156 10.1016/j.jde.2016.07.011 10.1007/s00526-016-1042-3 10.1007/s00526-018-1329-7 10.1016/j.anihpc.2006.01.003 10.1002/cpa.3160360405 10.1512/iumj.2009.58.3575 10.1007/978-1-4612-4146-1 |
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| References | Grigor’yan, A., Lin, Y., Yang, Y. Y.: Kazdan-Warner equation on graph. Calc. Var. Partial Differential Equations, 55(4), Art. 92, 13 pp. (2016) DingY HTanakaKMultiplicity of positive solutions of a nonlinear Schrödinger equationManuscripta Mathematica2003112109135200593310.1007/s00229-003-0397-x Grigor’yanALinYYangY YExistence of positive solutions to some nonlinear equations on locally finite graphsSci. China Math.20176013111324366580110.1007/s11425-016-0422-y Adams, R. A.: Sobolev Spaces, Academic Press, 1975 CaoD MNontrivial solution of semilinear equations with critical exponent in ℝ2Commun. Partial Differential Equations199217407435116343110.1080/03605309208820848 GeH BJiangW FYamabe equations on infinite graphsJ. Math. Anal. Appl.2018460885890375907610.1016/j.jmaa.2017.12.020 GeH BA p-th Yamabe equation on graphProceedings of the American Mathematical Society2018146522192224376737210.1090/proc/13929 LiY QWangZ QZengJGround states of nonlinear Schröodinger equations with potentialsAnn. Inst. H. Poincaré Anal. Non Linéaire200623829837227169510.1016/j.anihpc.2006.01.003 LêAEigenvalue problems for the p-LaplacianNonlinear Analysis20066410571099219681110.1016/j.na.2005.05.056 BrézisHNirenberg, L., Positive solutions of nonlinear elliptic equations involving critical Sobolev exponentsComm. Pure Appl. Math.19833643747770964410.1002/cpa.3160360405 ChenLCoulhonTHuaBRiesz transforms for bounded Laplacians on graphsMath. Z.2020294397417405007110.1007/s00209-019-02253-5 HuangX POn uniqueness class for a heat equation on graphsJ. Math. Anal. Appl.2012393377388292168110.1016/j.jmaa.2012.04.026 ShaoMMaoAMultiplicity of solutions to Schröodinger-Poisson system with concave-convex nonlinearitiesApplied Mathematics Letters201883212218379569410.1016/j.aml.2018.04.005 HeX MZouW MExistence and concentration of ground states for Schroödinger-Poisson equations with critical growthJournal of Mathematical Physics20125311910.1063/1.3683156 Keller, M., Schwarz, M.: The Kazdan-Warner equation on canonically compactifiable graphs. Calc. Var. Partial Differential Equations, 57(2) Art. 70, 18 pp. (2018) Ding, G.: Introduction to Banach Spaces (in Chinese), Science Press, 1997 Grigor’yanALinYYangY YYamabe type equations on graphsJ. Differential Equations201626149244943354296310.1016/j.jde.2016.07.011 ChungY SLeeY SChungS YExtinction and positivity of the solutions of the heat equations with absorption on networksJ. Math. Anal. Appl.2011380642652279442010.1016/j.jmaa.2011.03.006 ZhaoLZhangNExistence of solutions for a higher order Kirchhoff type problem with exponetial critical growthNonlinear Anal.2016132214226343396310.1016/j.na.2015.11.014 Lin, Y., Wu, Y. T.: The existence and nonexistence of global solutions for a semilinear heat equation on graphs. Calc. Var. Partial Differential Equations, 56(4), Art. 102, 22 pp. (2017) HanXShaoMZhaoLExistence and convergence of solutions for nonlinear biharmonic equations on graphsJournal of Differential Equations2020268739363961405361010.1016/j.jde.2019.10.007 WillemMMinimax Theorems1996BostonBirkhaäuser10.1007/978-1-4612-4146-1 NehariZOn a class of nonlinear second-order differential equationsTrans. AMS19609510112311189810.1090/S0002-9947-1960-0111898-8 ZhangNZhaoLConvergence of ground state solutions for nonlinear Schrödinger equations on graphsSci. 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Y.: Kazdan-Warner equation on graph. Calc. Var. Partial Differential Equations, 55(4), Art. 92, 13 pp. (2016) – reference: ZhaoLChangY YMin-max level estimate for a singular quasilinear polyharmonic equation in ℝ2mJ. Differential Equations201325424342464301620910.1016/j.jde.2012.12.005 – reference: Lin, Y., Wu, Y. T.: The existence and nonexistence of global solutions for a semilinear heat equation on graphs. Calc. Var. Partial Differential Equations, 56(4), Art. 102, 22 pp. (2017) – reference: HanXShaoMZhaoLExistence and convergence of solutions for nonlinear biharmonic equations on graphsJournal of Differential Equations2020268739363961405361010.1016/j.jde.2019.10.007 – reference: ZhaoLZhangNExistence of solutions for a higher order Kirchhoff type problem with exponetial critical growthNonlinear Anal.2016132214226343396310.1016/j.na.2015.11.014 – reference: BrézisHNirenberg, L., Positive solutions of nonlinear elliptic equations involving critical Sobolev exponentsComm. Pure Appl. Math.19833643747770964410.1002/cpa.3160360405 – reference: Adams, R. A.: Sobolev Spaces, Academic Press, 1975 – reference: CaoD MNontrivial solution of semilinear equations with critical exponent in ℝ2Commun. Partial Differential Equations199217407435116343110.1080/03605309208820848 – reference: Grigor’yanALinYYangY YYamabe type equations on graphsJ. Differential Equations201626149244943354296310.1016/j.jde.2016.07.011 – reference: HuangX POn uniqueness class for a heat equation on graphsJ. Math. Anal. Appl.2012393377388292168110.1016/j.jmaa.2012.04.026 – reference: ChungY SLeeY SChungS YExtinction and positivity of the solutions of the heat equations with absorption on networksJ. Math. Anal. Appl.2011380642652279442010.1016/j.jmaa.2011.03.006 – reference: LiY QWangZ QZengJGround states of nonlinear Schröodinger equations with potentialsAnn. Inst. H. Poincaré Anal. Non Linéaire200623829837227169510.1016/j.anihpc.2006.01.003 – reference: NehariZOn a class of nonlinear second-order differential equationsTrans. AMS19609510112311189810.1090/S0002-9947-1960-0111898-8 – reference: Grigor’yanALinYYangY YExistence of positive solutions to some nonlinear equations on locally finite graphsSci. China Math.20176013111324366580110.1007/s11425-016-0422-y – reference: HeX MZouW MExistence and concentration of ground states for Schroödinger-Poisson equations with critical growthJournal of Mathematical Physics20125311910.1063/1.3683156 – reference: RabinowitzP HOn a class of nonlinear Schröodinger equationsZ. Angew. Math. Phys.199243270291116272810.1007/BF00946631 – reference: LêAEigenvalue problems for the p-LaplacianNonlinear Analysis20066410571099219681110.1016/j.na.2005.05.056 – reference: LinYWuY TOn-diagonal lower estimate of heat kernels on graphsJ. Math. Anal. 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| Snippet | This paper is mainly concerned with the following nonlinear
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)... This paper is mainly concerned with the following nonlinear p-Laplacian equation −Δp(x)+(λa(x)+1)∣u∣p−2(x)u(x)=f(x,u(x)),inV on a locally finite graph G = (V,... |
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| SubjectTerms | Dirichlet problem Graphs Laplace equation Mathematics Mathematics and Statistics Mountains |
| Title | p-Laplacian Equations on Locally Finite Graphs |
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