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The Ground State Solutions to a Class of Biharmonic Choquard Equations on Weighted Lattice Graphs

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Abstract

In this paper, we consider the biharmonic Choquard equation with the nonlocal term on the weighted lattice graph \({\mathbb {Z}}^N\), namely for any \(p>1\) and \(\alpha \in (0,\,N)\)

$$\begin{aligned} \Delta ^2u-\Delta u+V(x)u=\left( \sum _{y\in {\mathbb {Z}}^N,\,y\not =x}\frac{|u(y)|^p}{d(x,\,y)^{N-\alpha }}\right) |u|^{p-2}u, \end{aligned}$$

where \(\Delta ^2\) is the biharmonic operator, \(\Delta \) is the \(\mu \)-Laplacian, \(V:{\mathbb {Z}}^N\rightarrow {\mathbb {R}}\) is a function, and \(d(x,\,y)\) is the distance between x and y. If the potential V satisfies certain assumptions, using the method of Nehari manifold, we prove that for any \(p>(N+\alpha )/N\), there exists a ground state solution of the above-mentioned equation. Compared with the previous results, we adopt a new method to finding the ground state solution from mountain-pass solutions.

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Authors and Affiliations

  1. School of Mathematics, Renmin University of China, Beijing, 100872, China

    Yang Liu

  2. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China

    Mengjie Zhang

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  1. Yang Liu
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  2. Mengjie Zhang
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Correspondence to Mengjie Zhang.

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Communicated by Alireza Amini Harandi.

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Liu, Y., Zhang, M. The Ground State Solutions to a Class of Biharmonic Choquard Equations on Weighted Lattice Graphs. Bull. Iran. Math. Soc. 50, 12 (2024). https://doi.org/10.1007/s41980-023-00846-9

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  • DOI: https://doi.org/10.1007/s41980-023-00846-9

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