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2023, Volume 31, Issue 2: 812-839. doi: 10.3934/era.2023041
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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 $.


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