Trudinger-Moser and Adams type inequalities on weighted Sobolev spaces and applications

Abstract

This thesis studies inequalities and embeddings involving a class of Sobolev spaces with potential weights without assuming any boundary condition. Suppose a Dirichlet boundary condition, those spaces have been extensively studied due to their applicability in radial elliptic problems for operators in great generality which include the p-Laplacian and the k-Hessian operators. In the bounded domains case without any boundary condition, we show a sharp embedding into weighted Lebesgue space Lq θ which generalizes [13, Theorem 1.1] and [22, Theorem 1.1]. Also, we prove sharp Adams-Trudinger-Moser embedding under the full norm and sharp Adams inequality with the Navier boundary condition generalizing [22, Theorem 1.3]. As applications, we conclude that the associated elliptic equations with nonlinearities in both forms of polynomial and exponential growths admit nontrivial solutions. Supposing an unbounded domain, our results provide sharp embeddings into weighted Lebesgue spaces Lq θ and the existence and non-existence of the maximizers for their Trudinger-Moser type inequalities. We also sharpen the maximal integrability by “removing" terms from the exponential series while maintaining the continuity of the embedding. Moreover, we establish the second order Adams’ inequalities with the exact growth condition.