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

In this paper, we consider the biharmonic Choquard equation with the nonlocal term on the weighted lattice graph \({\mathbb {Z}}^N\) ${\mathbb{Z}}^N$ , namely for any \(p>1\) $p>1$ and \(\alpha \in (0,\,N)\) $\alpha \in (0,N)$ \(\begin{aligned} \Delta ^2u-\Delta u+V(x)u=\left( \sum _{y\in {\mathbb {Z}}^N,\,y\not =x}\frac{|u(y)|^p}{d(x,\,y)^{N-\alpha }}\right) |u|^{p-2}u, \end{aligned}\) $$\begin{array}{c}\hfill {\mathrm{\Delta }}^{2}u-\mathrm{\Delta }u+V(x)u=\left(\sum _{y\in {\mathbb{Z}}^N,y\ne x},\frac{{|u(y)|}^p}{d{(x,y)}^{N-\alpha }}\right){|u|}^{p-2}u,\end{array}$$ where \(\Delta ^2\) ${\mathrm{\Delta }}^2$ is the biharmonic operator, \(\Delta \) $\mathrm{\Delta }$ is the \(\mu \) $\mu$ -Laplacian, \(V:{\mathbb {Z}}^N\rightarrow {\mathbb {R}}\) $V:{\mathbb{Z}}^N\to \mathbb{R}$ is a function, and \(d(x,\,y)\) $d(x,y)$ 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 \(p>(N+\alpha )/N\) $p>(N+\alpha )/N$ , 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.