Ground states of biharmonic equations on lattice graphs

In this paper, we are concerned with the existence of ground states to the following biharmonic equation on the lattice graph Δ² u-Δ u+V(x)u=f(x, u), x ∈ ZN. The analysis is performed if the potential V and the reaction f are T-periodic in x, and the mapping u ↦ f(x, u)/ u is non-decreasing on R \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 ↦ f(u)/ u non-decreasing on R \0\, the ground states are also obtained for the above equation. Finally, we extend the main results on the lattice graph ZN to quasi-transitive graphs. In our analysis, the mappings u ↦ f(x, u)/ u or u ↦ f(u)/ u are only non-decreasing on R \0\, which allows to consider larger classes of nonlinearities in the reaction.