Existence results for a class of nonlinear degenerate Navier problems

In this paper we are interested in the existence of solutions for Navier problem associated with the degenerate nonlinear elliptic equations
\begin{eqnarray*} &&{\Delta}{\big[}{\omega}_1(x) {\vert{\Delta}u\vert}^{p-2}{\Delta}u + {\omega}_2(x) {\vert{\Delta}u\vert}^{q-2}{\Delta}u {\big]} -\sum_{j=1}^n D_j{\bigl[}{\omega}_3(x) {\mathcal{A}}_j(x, u, {\nabla}u){\bigr]}\\ && = f_0(x) - \sum_{j=1}^nD_jf_j(x), \ \ {\mathrm{in}} \ \ {\Omega} \end{eqnarray*}
in the setting of the weighted Sobolev spaces.