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

Abstract

In this paper, we study the fractional $p$-Laplacian Choquard equation $$(-\Delta {)}_p^{s}u+h(x)|u{|}^{p-2}u=\left(R_{\alpha }\ast F(u)\right)f(u)$$ on lattice graphs ${\mathbb{Z}}^d$, where $s\in (0,1)$, $p\ge 2$, $\alpha \in (0,d)$ 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.

17 pages