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Solutions to discrete nonlinear Kirchhoff-Choquard equations
Analysis of PDEs · Submitted 18 Apr 2024
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Under some suitable assumptions on $V$ and $f$, we prove the existence of nontrivial solutions and ground state solutions by variational methods.
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In this paper, we study the discrete Kirchhoff-Choquard equation $$ -\left(a+b \int_{\mathbb{Z}^3}|\nabla u|^{2} d \mu\right) \Delta u+V(x) u=\left(R_{\alpha} *F(u)\right)f(u),\quad x\in \mathbb{Z}^3, $$ where $a,\,b>0$ are constants, $R_{\alpha}$ is the Green's function of the discrete fractional Laplacian with $\alpha \in(0,3)$, which has no singularity but has same asymptotics as the Riesz potential. Under some suitable assumptions on $V$ and $f$, we prove the existence of nontrivial solutions and ground state solutions by variational methods.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | In this paper, we study the discrete Kirchhoff-Choquard equation $$ -\left(a+b \int_{\mathbb{Z}^3}|\nabla u|^{2} d \mu\right) \Delta u+V(x) u=\left(R_{\alpha} *F(u)\right)f(u),\quad x\in \mathbb{Z}^3, $$ where $a,\,b>0$ are constants, $R_{\alpha}$ is the Green's function of the discrete fractional Laplacian with $\alpha \in(0,3)$, which has no singularity but has same asymptotics as the Riesz potential. | background 0.50 | 0.36 | 0.04 | asserted 0.94 |
| s1 | Under some suitable assumptions on $V$ and $f$, we prove the existence of nontrivial solutions and ground state solutions by variational methods. | result 0.57 | 0.96 | 0.93 | asserted 1.00 |
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Original abstract
In this paper, we study the discrete Kirchhoff-Choquard equation $$ -\left(a+b \int_{\mathbb{Z}^3}|\nabla u|^{2} d \mu\right) \Delta u+V(x) u=\left(R_{\alpha} *F(u)\right)f(u),\quad x\in \mathbb{Z}^3, $$ where $a,\,b>0$ are constants, $R_{\alpha}$ is the Green's function of the discrete fractional Laplacian with $\alpha \in(0,3)$, which has no singularity but has same asymptotics as the Riesz potential. Under some suitable assumptions on $V$ and $f$, we prove the existence of nontrivial solutions and ground state solutions by variational methods.
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For the discrete nonlinear Choquard equations, we refer the readers to [ref] .
The existence and convergence of solutions for the nonlinear Choquard equations on groups of polynomial growth · Introduction · 0.99
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