Search Advanced

Electronic Research Archive

2023, Volume 31, Issue 2: 812-839. doi: 10.3934/era.2023041
Previous Article Next Article
Research article Special Issues

Existence and asymptotical behavior of the ground state solution for the Choquard equation on lattice graphs

  • 1.
    Institute of Applied System Analysis, Jiangsu University, Zhenjiang 212013, China
  • 2.
    School of Mathematical Sciences, Jiangsu University, Zhenjiang 212013, China
  • Received: 29 September 2022 Revised: 14 November 2022 Accepted: 22 November 2022 Published: 01 December 2022

  • In this paper, we study the nonlinear Choquard equation

    $ \begin{equation*} - \Delta u + V(x)u = \left( {\sum\limits_{y \ne x \atop y \in { \mathbb {Z} ^{N}} } {\frac{|u(y)|^p}{|x-y|^{N-\alpha}}} }\right )|u|^{p-2}u \end{equation*} $

    on lattice graph $ \mathbb {Z}^{N} $. Under some suitable assumptions, we prove the existence of a ground state solution of the equation on the graph when the function $ V $ is periodic or confining. Moreover, when the potential function $ V(x) = \lambda a(x)+1 $ is confining, we obtain the asymptotic properties of the solution $ u_\lambda $ which converges to a solution of a corresponding Dirichlet problem as $ \lambda\rightarrow \infty $.

    Citation: Jun Wang, Yanni Zhu, Kun Wang. Existence and asymptotical behavior of the ground state solution for the Choquard equation on lattice graphs[J]. Electronic Research Archive, 2023, 31(2): 812-839. doi: 10.3934/era.2023041

    Related Papers:

  • In this paper, we study the nonlinear Choquard equation

    $ \begin{equation*} - \Delta u + V(x)u = \left( {\sum\limits_{y \ne x \atop y \in { \mathbb {Z} ^{N}} } {\frac{|u(y)|^p}{|x-y|^{N-\alpha}}} }\right )|u|^{p-2}u \end{equation*} $

    on lattice graph $ \mathbb {Z}^{N} $. Under some suitable assumptions, we prove the existence of a ground state solution of the equation on the graph when the function $ V $ is periodic or confining. Moreover, when the potential function $ V(x) = \lambda a(x)+1 $ is confining, we obtain the asymptotic properties of the solution $ u_\lambda $ which converges to a solution of a corresponding Dirichlet problem as $ \lambda\rightarrow \infty $.


    加载中


    [1] E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Stud. Appl. Math., 57 (1977), 93–105. https://doi.org/10.1002/sapm197757293 doi: 10.1002/sapm197757293
    [2] P. L. Lions, Some remarks on Hartree equation, Nonlinear Anal. Theory Methods Appl., 5 (1981), 1245–1256. https://doi.org/10.1016/0362-546X(81)90016-X doi: 10.1016/0362-546X(81)90016-X
    [3] V. Moroz, J. Van Schaftingen, Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics, J. Funct. Anal., 265 (2013), 153–184. https://doi.org/10.1016/j.jfa.2013.04.007 doi: 10.1016/j.jfa.2013.04.007
    [4] V. Moroz, J. Van Schaftingen, Groundstates of nonlinear Choquard equations: Hardy-Littlewood-Sobolev critical exponent, Commun. Contemp. Math., 17 (2015), 1550005. https://doi.org/10.1142/S0219199715500054 doi: 10.1142/S0219199715500054
    [5] V. Moroz, J. Van Schaftingen, Existence of groundstates for a class of nonlinear Choquard equations, Trans. Am. Math. Soc., 367 (2015), 6557–6579. https://doi.org/10.1090/S0002-9947-2014-06289-2 doi: 10.1090/S0002-9947-2014-06289-2
    [6] V. Moroz, J. Van Schaftingen, A guide to the Choquard equation, J. Fixed Point Theory Appl., 19 (2017), 773–813. https://doi.org/10.1007/s11784-016-0373-1 doi: 10.1007/s11784-016-0373-1
    [7] S. I. Pekar, Untersuchung über die Elektronentheorie der Kristalle, Akademie Verlag, Berlin, 1954. https://doi.org/10.1515/9783112649305
    [8] R. Penrose, On gravity's role in quantum state reduction, Gen. Relativ. Gravitation, 28 (1996), 581–600. https://doi.org/10.1007/BF02105068 doi: 10.1007/BF02105068
    [9] A. Elgart, B. Schlein, Mean field dynamics of Boson stars, Commun. Pure Appl. Math., 60 (2007), 500–545. https://doi.org/10.1002/cpa.20134 doi: 10.1002/cpa.20134
    [10] N. Ackermann, On a periodic Schrödinger equation with nonlocal superlinear part, Math. Z., 248 (2004), 423–443. https://doi.org/10.1007/s00209-004-0663-y doi: 10.1007/s00209-004-0663-y
    [11] C. O. Alves, A. B. Nóbrega, M. B. Yang, Multi-bump solutions for Choquard equation with deepening potential well, Calc. Var. Partial Differ. Equations, 55 (2016). https://doi.org/10.1007/s00526-016-0984-9 doi: 10.1007/s00526-016-0984-9
    [12] V. Moroz, J. Van Schaftingen, Semi-classical states for the Choquard equation, Calc. Var. Partial Differ. Equations, 52 (2015), 199–235. https://doi.org/10.1007/s00526-014-0709-x doi: 10.1007/s00526-014-0709-x
    [13] L. Guo, T. X. Hu, S. J. Peng, W. Shuai, Existence and uniqueness of solutions for Choquard equation involving Hardy-Littlewood-Sobolev critical exponent, Calc. Var. Partial Differ. Equations, 58 (2019), 128. https://doi.org/10.1007/s00526-019-1585-1 doi: 10.1007/s00526-019-1585-1
    [14] P. L. Lions, Solutions of Hartree-Fock equations for Coulomb systems, Commun. Math. Phys., 109 (1987), 33–97. https://doi.org/10.1007/BF01205672 doi: 10.1007/BF01205672
    [15] J. Wang, Existence of normalized solutions for the coupled Hartree-Fock type system, Math. Nachr., 294 (2021), 1987–2020. https://doi.org/10.1002/mana.201900230 doi: 10.1002/mana.201900230
    [16] J. Wang, Q. P. Geng, M. C. Zhu, Existence of the normalized solutions to the nonlocal elliptic system with partial confinement, Discrete Contin. Dyn. Syst., 39 (2019), 2187–2201. https://doi.org/10.3934/dcds.2019092 doi: 10.3934/dcds.2019092
    [17] J. K. Xia, X. Zhang, Saddle solutions for the critical Choquard equation, Calc. Var. Partial Differ. Equations, 60 (2021), 53. https://doi.org/10.1007/s00526-021-01919-5 doi: 10.1007/s00526-021-01919-5
    [18] I. M. Moroz, R. Penrose, P. Tod, Spherically-symmetric solutions of the Schrödinger-Newton equations, Classical Quantum Gravity, 15 (1998), 2733–2742. https://doi.org/10.1088/0264-9381/15/9/019 doi: 10.1088/0264-9381/15/9/019
    [19] A. Grigor'yan, Y. Lin, Y. Y. Yang, Yamabe type equations on graphs, J. Differ. Equations, 261 (2016), 4924–4943. https://doi.org/10.1016/j.jde.2016.07.011 doi: 10.1016/j.jde.2016.07.011
    [20] A. Grigor'yan, Y. Lin, Y. Y. Yang, Kazdan-Warner equation on graph, Calc. Var. Partial Differ. Equations, 55 (2016), 92. https://doi.org/10.1007/s00526-016-1042-3 doi: 10.1007/s00526-016-1042-3
    [21] A. Grigor'yan, Y. Lin, Y. Y. Yang, Existence of positive solutions to some nonlinear equations on locally finite graphs, Sci. China Math., 60 (2017), 1311–1324. https://doi.org/10.1007/s11425-016-0422-y doi: 10.1007/s11425-016-0422-y
    [22] X. Han, M. Shao, L. Zhao, Existence and convergence of solutions for nonlinear biharmonic equations on graphs, J. Differ. Equations, 268 (2020), 3936–3961. https://doi.org/10.1016/j.jde.2019.10.007 doi: 10.1016/j.jde.2019.10.007
    [23] H. Y. Huang, J. Wang, W. Yang, Mean field equation and relativistic Abelian Chern-Simons model on finite graphs, J. Funct. Anal., 281 (2021), 109218. https://doi.org/10.1016/j.jfa.2021.109218 doi: 10.1016/j.jfa.2021.109218
    [24] B. Hua, W. Xu, Existence of ground state solutions to some Nonlinear Schrödinger equations on lattice graphs, preprint, arXiv: 2108.00711.
    [25] Y. Lin, Y. T. Wu, On-diagonal lower estimate of heat kernels on graphs, J. Math. Anal. Appl., 456 (2017), 1040–1048. https://doi.org/10.1016/j.jmaa.2017.07.028 doi: 10.1016/j.jmaa.2017.07.028
    [26] Y. Lin, Y. T. Wu, The existence and nonexistence of global solutions for a semilinear heat equation on graphs, Calc. Var. Partial Differ. Equations, 56 (2017), 102. https://doi.org/10.1007/s00526-017-1204-y doi: 10.1007/s00526-017-1204-y
    [27] N. Zhang, L. Zhao, Convergence of ground state solutions for nonlinear Schrödinger equations on graphs, Sci. China Math., 61 (2018), 1481–1494. https://doi.org/10.1007/s11425-017-9254-7 doi: 10.1007/s11425-017-9254-7
    [28] A. Szulkin, T. Weth, The method of Nehari manifold, 2010. Available from: https://staff.math.su.se/andrzejs/publications/Nehari.pdf
    [29] H. Ge, Kazdan-Warner equation on graph in the negative case, J. Math. Anal. Appl., 453 (2017), 1022–1027. https://doi.org/10.1016/j.jmaa.2017.04.052 doi: 10.1016/j.jmaa.2017.04.052
    [30] A. Huang, Y. Lin, S. Yau, Existence of solutions to mean field equations on graphs, Commun. Math. Phys., 377 (2020), 613–621. https://doi.org/10.1007/s00220-020-03708-1 doi: 10.1007/s00220-020-03708-1
    [31] M. Keller, M. Schwarz, The Kazdan-Warner equation on canonically compactifiable graphs, Calc. Var. Partial Differ. Equations, 57 (2018), 70. https://doi.org/10.1007/s00526-018-1329-7 doi: 10.1007/s00526-018-1329-7
    [32] L. Sun, J. Zhu, Global existence and convergence of a flow to Kazdan-Warner equation with non-negative prescribed function, Calc. Var. Partial Differ. Equations, 60 (2021), 42. https://doi.org/10.1007/s00526-020-01873-8 doi: 10.1007/s00526-020-01873-8
    [33] Y. Lin, Y. T. Wu, Blow-up problems for nonlinear parabolic equations on locally finite graphs, Acta Math. Sci., 38 (2018), 843–856. https://doi.org/10.1016/S0252-9602(18)30788-4 doi: 10.1016/S0252-9602(18)30788-4
    [34] Y. Lin, Y. Y. Yang, A heat flow for the mean field equation on a finite graph, Calc. Var. Partial Differ. Equations, 60 (2021), 206. https://doi.org/10.1007/s00526-021-02086-3 doi: 10.1007/s00526-021-02086-3
    [35] P. L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case, Part Ⅰ, Ann. Inst. Henri Poincare C, 1 (1984), 109–145. https://doi.org/10.1016/S0294-1449(16)30428-0 doi: 10.1016/S0294-1449(16)30428-0
    [36] P. L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case, Part Ⅱ, Ann. Inst. Henri Poincare C, 1 (1984), 223–283. https://doi.org/10.1016/S0294-1449(16)30422-X doi: 10.1016/S0294-1449(16)30422-X
    [37] T. Bartsch, Z. Q. Wang, Multiple positive solutions for a nonlinear Schrödinger equation, Z. Angew. Math. Phys., 51 (2000), 366–384. https://doi.org/10.1007/PL00001511 doi: 10.1007/PL00001511
    [38] B. Hua, R. Li, The existence of extremal functions for discrete Sobolev inequalities on lattice graphs, J. Differ. Equations, 305 (2021), 224–241. https://doi.org/10.1016/j.jde.2021.10.016 doi: 10.1016/j.jde.2021.10.016
    [39] M. Willem, Minimax Theorems, Birkhäuser, Boston, MA, 1996. https://doi.org/10.1007/978-1-4612-4146-1
    [40] H. Brézis, E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Am. Math. Soc., 88 (1983), 486–490. https://doi.org/10.2307/2044999 doi: 10.2307/2044999
    [41] G. G. Huang, C. M. Li, X. M. Yin, Existence of the maximizing pair for the discrete Hardy-Littlewood-Sobolev inequality, Discrete Contin. Dyn. Syst., 35 (2015), 935–942. https://doi.org/10.3934/dcds.2015.35.935 doi: 10.3934/dcds.2015.35.935
    [42] K. C. Chang, Methods in Nonlinear Analysis, Springer-Verlag, Berlin, 2005. Available from: https://link.springer.com/book/10.1007/3-540-29232-2.
  • Reader Comments
  • © 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索
Electronic Research Archive
1 1.3

Metrics

Article views(1269) PDF downloads(111) Cited by(4)

Preview PDF
Download XML
Export Citation
Article outline

Other Articles By Authors

Related pages

Tools

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog