Homoclinic solutions for discrete fractional p-Laplacian equation via the Nehari manifold method

In the present paper, we study by the variational approach the multiplicity of homoclinic solutions for the next discrete fractional p-Laplacian equation \(\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta _{\mathcal {G}})^{s}_{p}u(\xi )+V(\xi )\vert u(\xi )\vert ^{p-2}u(\xi ) &{} \text {for }\xi \in \mathbb {Z}, \\ \ \ \ \ =f(\xi , u(\xi ))+\lambda h(\xi )\vert u(\xi )\vert ^{q-2}u(\xi )\\ u(\xi )\longrightarrow 0 &{} \text {as }\vert \xi \vert \longrightarrow \infty \end{array}\right. } \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{(-{\mathrm{\Delta }}_{\mathcal{G}})}_p^{s}u(\xi )+V(\xi ){|u(\xi )|}^{p-2}u(\xi )\hfill & \text{for}\xi \in \mathbb{Z},\hfill \\ =f(\xi ,u(\xi ))+{\lambda h(\xi )|u(\xi )|}^{q-2}u(\xi )\hfill \\ u(\xi )⟶0\hfill & \text{as}|\xi |⟶\infty \hfill \end{array}\right)\end{array}$$ where \(s\in (0,1)\) $s\in (0,1)$ , \(2<r<p<q\) $2<r<p<q$ , \(V: \mathbb {Z}\longmapsto \mathbb {R}^{+}\) $V:\mathbb{Z}⟼{\mathbb{R}}^{+}$ , \(\lambda \) $\lambda$ is a positive parameter, \(h \in C(\mathbb {Z},\mathbb {R})\) $h\in C(\mathbb{Z},\mathbb{R})$ , and \(f\in C(\mathbb {Z}\times \mathbb {R},\mathbb {R})\) $f\in C(\mathbb{Z}\times \mathbb{R},\mathbb{R})$ . Upon appropriate assumptions on the potential V and nonlinearity f, we use the Nehari manifold in conjunction with fibering maps to prove the multiple homoclinic solutions for a discrete fractional p-Laplacian equation. Furthermore, we provide an illustrative example to showcase our approach in a particular case.