Ground state solutions of $p$-Laplacian equations with nonnegative potentials on Lattice 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 $p\ge 2$, 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.

arXiv comment31 pages