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Abstract: We prove an extension of the Stein–Weiss weighted estimates for fractional integrals, in the context of spaces with different integrability properties in the radial and the angular direction. In this way, the classical estimates can be unified with their improved radial versions. A number of consequences are obtained: in particular we deduce precised versions of weighted Sobolev embeddings, Caffarelli–Kohn–Nirenberg estimates, and Strichartz estimates for the wave equation, which extend the radial improvements to the case of arbitrary functions. [Copyright &y& Elsevier]

Abstract: We prove the existence of nontrivial critical points of the functional related to the Caffarelli–Kohn–Nirenberg inequality and its higher order variant by Lin. As a consequence we obtain nontrivial solutions of the degenerate elliptic equation We also show that when , has infinitely many critical points. [Copyright &y& Elsevier]

Abstract: Some multiplicity results are presented for the eigenvalue problem where is an open bounded domain with smooth boundary, , , , and is sublinear at infinity and superlinear at the origin. Various cases are treated depending on the behaviour of the nonlinear term g. [Copyright &y& Elsevier]

Abstract: In this note we provide simple and short proofs for a class of inequalities of Caffarelli–Kohn–Nirenberg type with sharp constants. Our approach suggests some definitions of weighted Sobolev spaces and their embedding into weighted spaces. These may be useful in studying solvability of problems involving new singular PDEs. [Copyright &y& Elsevier]

In this article we study the existence and non-existence of extremals for the following family of Hardy-Sobolev inequalities (∫ (R +) k × R N − k | x B u | q) 1 q ≤ C (∫ (R +) k × R N − k | x A ∇ u | p) 1 p , which holds for suitable values of A , B ∈ R N , q > p > 1. Here the quantity x A (respectively x B) denotes the monomial weight defined as x A = | x 1 | a 1 ⋅ ... ⋅ | x k | a k (respectively x B = | x 1 | b 1 ⋅ ... ⋅ | x k | b k ). [ABSTRACT FROM AUTHOR]