  We establish embeddings on a class of Sobolev spaces with potential weights
on unbounded domains. Our results provide embeddings into weighted Lebesgue
spaces $L^q_\theta$ with radial power weights and establish the existence and
non-existence of the maximizers for their Trudinger-Moser type inequalities. We
also sharpen the maximal integrability by ``removing" necessary terms from the
exponential series while maintaining the continuity of the embedding.
