Existence and convergence of ground state solutions for Choquard-type systems on lattice graphs

Abstract

In this paper, we study the $p$-Laplacian system with Choquard-type nonlinearity $$\{\begin{array}{l}-{\Delta }_{p}u+(\lambda a+1)|u{|}^{p-2}u=\frac{1}{\gamma }\left(R_{\alpha }\ast F(u,v)\right)F_u(u,v),\\ -{\Delta }_{p}v+(\lambda b+1)|v{|}^{p-2}v=\frac{1}{\gamma }\left(R_{\alpha }\ast F(u,v)\right)F_v(u,v),\end{array}$$ on lattice graphs ${\mathbb{Z}}^N$, where $\alpha \in (0,N),p\ge 2,\gamma >\frac{(N+\alpha )p}{2N},\lambda >0$ is a parameter and $R_{\alpha }$ is the Green's function of the discrete fractional Laplacian that behaves as the Riesz potential. Under some assumptions on the functions $a,b$ and $F$, we prove the existence and asymptotic behavior of ground state solutions by the method of Nehari manifold.

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