The existence of solutions to the nonhomogeneous degenerate nonlinear elliptic equations

In this paper we are interested in the existence and uniqueness of solutions for Dirichlet problem associated with the degenerate nonlinear elliptic equations \begin{align*}&- \mathrm{div}\big[{\mathcal{A}}(x, {\nabla}u)\, {\omega}_2(x) + {\mathcal{B}}(x, {\nabla}u)\, {\nu}_1(x)\big] + {\mathcal{H}}(x,u,{\nabla}u){\nu}_2 + {\vert u \vert}^{p-2}u\,{\omega}_1\\ &= {\rho}_0 - \sum_{j=1}^nD_j{\rho}_j,\\ & u - {\psi}\, {\in}\, W_0^{1,p}(\Omega , {\omega}_1 , {\omega}_2),\end{align*} in the setting of the weighted Sobolev spaces.