Existence and asymptotical behavior of the ground state solution for the Choquard equation on lattice graphs

Abstract

<abstract><p>In this paper, we study the nonlinear Choquard equation</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} - \Delta u + V(x)u = \left( {\sum\limits_{y \ne x \atop y \in { \mathbb {Z} ^{N}} } {\frac{|u(y)|^p}{|x-y|^{N-\alpha}}} }\right )|u|^{p-2}u \end{equation*} $\end{document} </tex-math></disp-formula></p> <p>on lattice graph $ \mathbb {Z}^{N} $. Under some suitable assumptions, we prove the existence of a ground state solution of the equation on the graph when the function $ V $ is periodic or confining. Moreover, when the potential function $ V(x) = \lambda a(x)+1 $ is confining, we obtain the asymptotic properties of the solution $ u_\lambda $ which converges to a solution of a corresponding Dirichlet problem as $ \lambda\rightarrow \infty $.</p></abstract>