In this article, we study the biharmonic elliptic problem with the secondnd Hessian $$\displaylines{ \Delta^2u =S_2(D^2u)+\lambda f(x) |u|^{p-1}u,\quad \text{in } \Omega \subset \mathbb{R}^3, \cr u =\frac{\partial u}{\partial n}=0, \quad \text{on } \partial\Omega, }$$ where $f(x)\in C(\bar{\Omega})$ is a sign-changing weight function. By using variational methods and some properties of the Nehari manifold, we prove that the biharmonic elliptic problem has at least two nontrivial solutions.