Existence of positive and nonnegative eigenfunctions for a fourth order operator with definite and indefinite weights

In this paper, we study the existence of solutions for the following eigenvalue problem: $\left(\mathrm{LP}\right)\left(\begin{array}{ccc}(-\Delta +d_1)(-\Delta +d_2)u+m\left(x\right)u=\lambda a\left(x\right)u& & \text{in}\Omega \\ u⁄\equiv 0,u\ge 0& & \text{in}\Omega \\ \Delta u=u=0& & \text{on}\partial \Omega \end{array}\right)$ where $\Omega \subset R^N$ is a smooth bounded domain, $d_1,d_2\in R$ and $a\left(\cdot \right),m\left(\cdot \right)\in L^{\infty }\left(\Omega \right)$ may have indefinite sign.