Electronic Research Archive (Jan 2023)
Existence and asymptotical behavior of the ground state solution for the Choquard equation on lattice graphs
Abstract
In this paper, we study the nonlinear Choquard equation $ \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*} $ 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 $.
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