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

IF 0.7 4区数学 Q2 MATHEMATICS

Bulletin of The Iranian Mathematical Society Pub Date : 2024-01-31 DOI:10.1007/s41980-023-00846-9

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引用次数: 0

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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加权网格图上一类双谐周夸方程的基态解

摘要本文考虑了加权网格图 ${\mathbb {Z}}^N$ 上带有非局部项的双谐波 Choquard 方程，即对于任意 $p>1$ 和 $\alpha \in (0,\,N)$ $$begin{aligned}。\Delta ^2u-\Delta u+V(x)u=\left( \sum _{y\in {\mathbb {Z}}^N,\,y\not =x}\frac{|u(y)|^p}{d(x,\,y)^{N-\alpha }}\right) ||u|^{p-2}u、\end{aligned}$$ 其中 $\Delta ^2$是双谐波算子, $\Delta $是拉普拉奇算子, $V. {\mathbb {Z}) 是拉普拉奇算子：{是一个函数，而（d(x,\,y)）是x和y之间的距离。如果势 V 满足一定的假设条件，使用奈哈里流形的方法，我们可以证明对于任何 \(p>(N+\alpha )/N$都存在上述方程的基态解。与之前的结果相比，我们采用了一种新的方法，从山形通解中找到基态解。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Bulletin of The Iranian Mathematical Society Mathematics-General Mathematics

CiteScore

1.40

自引率

0.00%

发文量

期刊介绍： The Bulletin of the Iranian Mathematical Society (BIMS) publishes original research papers as well as survey articles on a variety of hot topics from distinguished mathematicians. Research papers presented comprise of innovative contributions while expository survey articles feature important results that appeal to a broad audience. Articles are expected to address active research topics and are required to cite existing (including recent) relevant literature appropriately. Papers are critically reviewed on the basis of quality in its exposition, brevity, potential applications, motivation, value and originality of the results. The BIMS takes a high standard policy against any type plagiarism. The editorial board is devoted to solicit expert referees for a fast and unbiased review process.