This work concerns the shape optimization problem governed by some elliptic equations in exterior domains. The existence of optimal solutions is obtained and the so-called. G-property for a family O of open subsets is established; i.e., if Omega(m), Omega -> O are such that Omega(m) -> Omega in the Hausdorff metric, then for any K subset of subset of R-N\ Omega, there exists m(K) > 0 such that for all m >= m(K), K subset of subset of R-N\Omega(m).