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

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.