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We derive sharp Sobolev embeddings on a class of Sobolev spaces withpotential weights without assuming any boundary conditions. Moreover, weconsider the Adams-type inequalities for the borderline Sobolev embedding intothe exponential class with a sharp constant. As applications, we prove that theassociated elliptic equations with nonlinearities in both forms of polynomialand exponential growths admit nontrivial solutions.

Sharp Sobolev and Adams-Trudinger-Moser embeddings on weighted Sobolev spaces and their applications