Trudinger-Moser embeddings on weighted Sobolev spaces on unbounded domains

We establish embeddings on a class of Sobolev spaces with potential weights on unbounded domains. Our results provide embeddings into weighted Lebesgue spaces Lq_θ 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.