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

Trudinger-Moser embeddings on weighted Sobolev spaces on unbounded domains