The authors extend results from [\textit{G. DalMaso}, \textit{V. Mosco} and \textit{M. A. Vivaldi}, Acta Math. 163, No. 1/2, 57-107 (1989; Zbl 0696.49016)] to double obstacle problems \[ u\in H^ 1 (\Omega, w)\cap L^ \infty (\Omega), \quad \psi_ 1\leq u\leq \psi_ 2, \qquad {\mathbf a} (u,v- u)+ \int_ \Omega H(x,u, \nabla u) (v-u) dx\geq 0 \] \[ \text{for all } v\in H^ 1 (\Omega,w)\cap L^ \infty (\Omega), \qquad \psi_ 1\leq v\leq \psi_ 2, \] where the weight function \(w\) is from Muckenhoupt's \(A_ 2\) class and the principal part of the operator is linear such that \(a_{ij} \xi_ i\cdot \xi_ j \approx w(x) |\xi |^ 2\).