Existence of solutions for a degenerate auasilinear elliptic system in bounded domain

Using variational methods, we study the existence of weak solutions forthe degenerate quasilinear elliptic system$$\left\{\begin{array}{ll}- \mathrm{div}\Big(h_1(x)|\nabla u|^{p-2}\nabla u\Big) = F_{u}(x,u,v) &\text{ in } \Omega,\\-\mathrm{div}\Big(h_2(x)|\nabla v|^{q-2}\nabla v\Big) = F_{v}(x,u,v) &\text{ in } \Omega,\\u=v=0 & \textrm{ on } \partial\Omega,\end{array}\right.$$where $\Omega\subset \mathbb R^N$ is a smooth bounded domain, $\nabla F= (F_u,F_v)$ stands for the gradient of $C^1$-function $F:\Omega\times\mathbb R^2 \to \mathbb R$, the weights $h_i$, $i=1,2$ are allowed to vanish somewhere,the primitive $F(x,u,v)$ is intimately related to the first eigenvalue of acorresponding quasilinear system.