$G$-compactness of sequences of non-linear operators of Dirichlet problems with a variable domain of definition

For a sequence of operators $A_s\colon\overset{\circ}{W}{}^{1,m}(\Omega_s)\to\bigl(\overset{\circ}{W}{}^{1,m}(\Omega_s)\bigr)^*$ in divergence form we prove a theorem concerning the choice of a subsequence that $G$-converges to the operator $\widehat A\colon\overset{\circ}{W}{}^{1,m}(\Omega)\to\bigl(\overset{\circ}{W}{}^{1,m}(\Omega)\bigr)^*$ with the same leading coefficients as the operator $A_s$ and some additional lower coefficient $b(x,u)$. We give a procedure for constructing the function $b(x,u)$. We discuss the question of whether the principal condition under which the choice theorem is established is necessary. We prove criteria for this condition to hold.