On existence results for a class of biharmonic elliptic problems without (AR) condition

In this paper, we study the following biharmonic elliptic equation in $ \mathbb{R}^{N} $: \begin{document}$ \Delta^{2}\psi-\Delta \psi+P(x)\psi = g(x, \psi), \ \ x\in\mathbb{R}^{N}, $\end{document} where $ g $ and $ P $ are periodic in $ x_{1}, \cdots, x_{N} $, $ g(x, \psi) $ is subcritical and odd in $ \psi $. Without assuming the Ambrosetti-Rabinowitz condition, we prove the existence of infinitely many geometrically distinct solutions for this equation, and the existence of ground state solutions is established as well.