Ground states of biharmonic equations on lattice graphs

Abstract

In this paper, we are concerned with the existence of ground states to the following biharmonic equation on the lattice graph $$ \Delta^2 u-\Delta u+V(x)u=f(x, u), \quad x \in \mathbb{Z}^N. $$ The analysis is performed if the potential $V$ and the reaction $f$ are $T$-periodic in $x$, and the mapping $u \mapsto {f(x, u)}/{\vert u\vert}$ is non-decreasing on $\mathbb{R}\setminus \{0\}$. By using the variational methods, we establish the existence of ground states for the above problem. Moreover, if the potential $V$ has a bounded potential well and $f(x, u)=f(u)$ with $u \mapsto {f(u)}/{\vert u\vert}$ non-decreasing on $\mathbb{R}\setminus \{0\}$, the ground states are also obtained for the above equation. Finally, we extend the main results on the lattice graph $\mathbb{Z}^N$ to quasi-transitive graphs. In our analysis, the mappings $u \mapsto {f(x, u)}/{\vert u\vert}$ or $u \mapsto {f(u)}/{\vert u\vert}$ are only non-decreasing on $\mathbb{R}\setminus \{0\}$, which allows to consider larger classes of nonlinearities in the reaction.