In this paper, we consider the quasi-linear elliptic problem

$$-M\left( {{\int }_{{{\mathbb{R}}^{N}}}}{{\left| x \right|}^{-ap}}{{\left| {{\nabla }_{u}} \right|}^{p}}dx \right)\,\text{div}\left( {{\left| x \right|}^{-ap}}{{\left| \nabla u \right|}^{p-2}}\nabla u \right)=\frac{\alpha }{\alpha +\beta }H\left( x \right){{\left| u \right|}^{\alpha -2}}u{{\left| v \right|}^{\beta }}+\text{ }\lambda \text{ }{{\text{h}}_{1}}\left( x \right){{\left| u \right|}^{q-2}}u,$$

$$-M\left( {{\int }_{{{\mathbb{R}}^{N}}}}{{\left| x \right|}^{-ap}}{{\left| \nabla v \right|}^{p}}dx \right)\,\text{div}\left( {{\left| x \right|}^{-ap}}{{\left| \nabla v \right|}^{p-2}}\nabla v \right)=\frac{\beta }{\alpha +\beta }H\left( x \right){{\left| v \right|}^{\beta -2}}v{{\left| u \right|}^{\alpha }}+\mu {{h}_{2}}\left( x \right){{\left| v \right|}^{q-2}}v,$$

$$u\left( x \right)>0,v\left( x \right)>0,x\in {{\mathbb{R}}^{N}},$$

where $\text{ }\lambda \text{ ,}\mu >\text{0,}\text{1}<\text{p}<\text{N,}\text{1}<\text{q}<\text{p}<\text{p}\left( \tau +1 \right)<\alpha +\beta <{{p}^{*}}=\frac{{{N}_{p}}}{N-p},0\le a<\frac{N-p}{p},a\le b<a+1,d=a+1-b>0,M\left( s \right)=k+l{{s}^{\tau }},k>0,l,\tau \ge 0$ and the weight $H\left( x \right),\,{{h}_{1}}\left( x \right),\,{{h}_{2}}\left( x \right)$ are continuous functions that change sign in ${{\mathbb{R}}^{N}}$ . We will prove that the problem has at least two positive solutions by using the Nehari manifold and the fibering maps associated with the Euler functional for this problem.

We consider the weak closure WZ of the set Z of all feasible pairs (solution, flow) of the family of potential elliptic systems $$ \begin{array}{c}\mbox{div}\left(\sum\limits_{s=1}^{s_0}\sigma_s(x)F_s^\prime (\nabla u(x)+g(x))-f(x)\right)=0\;\mbox{in}\,\Omega, u=(u_1,\dots, u_m)\in H_0^1(\Omega;{\bf R}^m),\;\sigma=(\sigma_1,\dots,\sigma_{s_0})\in S,\end{array} $$ where Ω ⊂ Rn is a bounded Lipschitz domain, Fs are strictly convex smooth functions with quadratic growth and $S=\{\sigma\, measurable\,\mid\,\sigma_s(x)=0\;\mbox{or}\,1,\;s=1,\dots,s_0,\;\sigma_1(x)+\cdots +\sigma_{s_0}(x)=1\}$. We show that WZ is the zero level set for an integral functional with the integrand $Q\cal F$ being the A-quasiconvex envelope for a certain function $\cal F$ and the operator A = (curl,div)m. If the functions Fs are isotropic, then on the characteristic cone Λ (defined by the operator A) $Q{\cal F}$ coincides with the A-polyconvex envelope of $\cal F$ and can be computed by means of rank-one laminates.