Ground state solutions of $p$-Laplacian equations with nonnegative potentials on Lattice graphs

TL;DR

This paper investigates ground state solutions of the $p$-Laplacian equation on lattice graphs with nonnegative potentials, establishing existence and multiplicity results, and extending findings to Cayley graphs.

Contribution

It introduces new existence and multiplicity results for $p$-Laplacian equations on lattice and Cayley graphs using the Nehari manifold method.

Findings

Existence of ground state solutions under certain growth conditions.

Infinitely many solutions when $f$ is odd and $p\,\geq 2$.

Extension of results from $\,\mathbb{Z}^N$ to Cayley graphs.

Abstract

In this paper, we study the $p$-Laplacian equation $$ -\Delta_p u + V(x)|u|^{p-2}u = f(x,u) $$ on the lattice graph $\mathbb{Z}^N$ with nonnegative potentials, where $\Delta_p$ is the discrete $p$-Laplacian and $p\in(1,\infty)$. By employing the Nehari manifold method, we establish the existence of ground state solutions under suitable growth conditions on the nonlinearity $f(x,u)$, provided that the potential $V(x)$ is either periodic or bounded. Moreover, we prove that if $f$ is odd in $u$ and $p\geq2$, then the above equation admits infinitely many geometrically distinct solutions. Finally, we extend these results from $\mathbb{Z}^N$ to the more general setting of Cayley graphs.