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Volume 38, Issue 2
The Existence and Convergence of Solutions for the Nonlinear Choquard Equations on Groups of Polynomial Growth

Ruowei Li & Lidan Wang

DOI: 10.4208/jpde.v38.n2.7

J. Part. Diff. Eq., 38 (2025), pp. 227-250.

Published online: 2025-06

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  • Abstract

In this paper, we study the nonlinear Choquard equation $$\Delta^2u-\Delta u+(1+\lambda a(x))u=(R_{\alpha}*|u|^p)|u|^{p-2}u$$on a Cayley graph of a discrete group of polynomial growth with the homogeneous dimension $N ≥ 1,$ where $α ∈ (0,N),$ $p>\frac{N+α}{N},$ $λ$ is a positive parameter and $R_α$ stands for the Green’s function of the discrete fractional Laplacian, which has no singularity at the origin but has same asymptotics as the Riesz potential at infinity. Under some assumptions on $a(x),$ we establish the existence and asymptotic behavior of ground state solutions for the nonlinear Choquard equation by the method of Nehari manifold.

  • Keywords

Nonlinear Choquard equation, discrete Green’s function, ground state solutions, Cayley graphs.

  • AMS Subject Headings

35J35, 35J91, 35R02

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JPDE-38-227, author = {Li , Ruowei and Wang , Lidan}, title = {The Existence and Convergence of Solutions for the Nonlinear Choquard Equations on Groups of Polynomial Growth}, journal = {Journal of Partial Differential Equations}, year = {2025}, volume = {38}, number = {2}, pages = {227--250}, abstract = {

In this paper, we study the nonlinear Choquard equation $$\Delta^2u-\Delta u+(1+\lambda a(x))u=(R_{\alpha}*|u|^p)|u|^{p-2}u$$on a Cayley graph of a discrete group of polynomial growth with the homogeneous dimension $N ≥ 1,$ where $α ∈ (0,N),$ $p>\frac{N+α}{N},$ $λ$ is a positive parameter and $R_α$ stands for the Green’s function of the discrete fractional Laplacian, which has no singularity at the origin but has same asymptotics as the Riesz potential at infinity. Under some assumptions on $a(x),$ we establish the existence and asymptotic behavior of ground state solutions for the nonlinear Choquard equation by the method of Nehari manifold.

}, issn = {2079-732X}, doi = {https://doi.org/10.4208/jpde.v38.n2.7}, url = {http://global-sci.org/intro/article_detail/jpde/24218.html} }
TY - JOUR T1 - The Existence and Convergence of Solutions for the Nonlinear Choquard Equations on Groups of Polynomial Growth AU - Li , Ruowei AU - Wang , Lidan JO - Journal of Partial Differential Equations VL - 2 SP - 227 EP - 250 PY - 2025 DA - 2025/06 SN - 38 DO - http://doi.org/10.4208/jpde.v38.n2.7 UR - https://global-sci.org/intro/article_detail/jpde/24218.html KW - Nonlinear Choquard equation, discrete Green’s function, ground state solutions, Cayley graphs. AB -

In this paper, we study the nonlinear Choquard equation $$\Delta^2u-\Delta u+(1+\lambda a(x))u=(R_{\alpha}*|u|^p)|u|^{p-2}u$$on a Cayley graph of a discrete group of polynomial growth with the homogeneous dimension $N ≥ 1,$ where $α ∈ (0,N),$ $p>\frac{N+α}{N},$ $λ$ is a positive parameter and $R_α$ stands for the Green’s function of the discrete fractional Laplacian, which has no singularity at the origin but has same asymptotics as the Riesz potential at infinity. Under some assumptions on $a(x),$ we establish the existence and asymptotic behavior of ground state solutions for the nonlinear Choquard equation by the method of Nehari manifold.

Li , Ruowei and Wang , Lidan. (2025). The Existence and Convergence of Solutions for the Nonlinear Choquard Equations on Groups of Polynomial Growth. Journal of Partial Differential Equations. 38 (2). 227-250. doi:10.4208/jpde.v38.n2.7
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