In this paper, we study the following biharmonic elliptic equation in ${\mathbb{R}}^N$:

$${\mathrm{\Delta }}^2\psi -\mathrm{\Delta }\psi +P(x)\psi =g(x,\psi ),x\in {\mathbb{R}}^N,$$

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.