Positive solutions to fractional $p$-Laplacian Choquard equation on lattice graphs

Abstract

In this paper, we study the fractional $p$-Laplacian Choquard equation $$ (-\Delta)_{p}^{s} u+h(x)|u|^{p-2} u=\left(R_{\alpha} *F(u)\right)f(u) $$ on lattice graphs $\mathbb{Z}^d$, where $s\in(0,1)$, $ p\geq 2$, $\alpha \in(0, d)$ and $R_\alpha$ represents the Green's function of the discrete fractional Laplacian that behaves as the Riesz potential. Under suitable assumptions on the potential function $h$, we first prove the existence of a strictly positive solution by the mountain-pass theorem for the nonlinearity $f$ satisfying some growth conditions. Moreover, if we add some monotonicity condition, we establish the existence of a positive ground state solution by the method of Nehari manifold.

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