Existence and multiplicity results for a doubly anisotropic problem with sign-changing nonlinearity

We consider in this paper the following problem $ -\sum\limits_{i=1}^{N}\partial _{i}\left[ \left\vert \partial _{i}u\right\vert ^{p_{i}-2}\partial _{i}u\right] -\sum\limits_{i=1}^{N} \partial _{i}\left[ \left\vert \partial _{i}u\right\vert ^{q_{i}-2}\partial _{i}u\right] =\lambda f(u) \text{ \ }\ \text{in } \Omega,\\ $ $ u=0\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\ \text{on }\partial \Omega. $ Where $\Omega $ is a bounded regular domain in $\mathbb{R}^{N}$, $% 1 p_{1}\leq p_{2}\leq ...\leq p_{N}$ and $1 q_{1}\leq q_{2}\leq ...\leq q_{N},$ we will also assume that $f$ is a continuous function, that have a finite number of zeroes, changing sign between them.