Multiplicity of solutions of weighted (p,q)-Laplacian with small source

In this article, we study the existence of infinitely many solutions to the degenerate quasilinear elliptic system

$\begin{array}{rr}\hfill \begin{array}{r}\hfill -\text{div}\left(h_1\left(x\right){\left|?u\right|}^{p-2}?u\right)=d\left(x\right){\left|u\right|}^{r-2}u+G_u\left(x,u,\upsilon \right)\\ \hfill -\text{div}\left(h_2\left(x\right){\left|?\upsilon \right|}^{q-2}?\upsilon \right)=f\left(x\right){\left|\upsilon \right|}^{s-2}\upsilon +G_{\upsilon }\left(x,u,\upsilon \right)\\ \hfill \text{u}=\upsilon =\text{0}\end{array}& \hfill \begin{array}{l}\text{in}\Omega \hfill \\ \text{in}\Omega \hfill \\ \text{on}?\Omega \hfill \end{array}\end{array}$

where Ω is a bounded domain in ${\mathbb{R}}^N$ with smooth boundary ?Ω, N≥2, 1 < r < p < ∞, 1 < s < q < ∞; h₁(x) and h₂(x) are allowed to have “essential” zeroes at some points in $\Omega ;\text{d}\left(x\right){\left|u\right|}^{r-2}u$ and $f\left(x\right){\left|\upsilon \right|}^{s-2}\upsilon$ are small sources with G_u(x,u,v), G_v(x,u,v) being their high-order perturbations with respect to (u,v) near the origin, respectively.