Cocompactness and minimizers for inequalities of Hardy-Sobolev type involving N-Laplacian

Adimurthi, ; Marcos do O, Joao ; Tintarev, Kyril (2010) Cocompactness and minimizers for inequalities of Hardy-Sobolev type involving N-Laplacian Nonlinear Differential Equations and Applications, 17 (4). pp. 467-477. ISSN 1021-9722

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Official URL: http://www.springerlink.com/content/t22023r2j05237...

Related URL: http://dx.doi.org/10.1007/s00030-010-0063-4

Abstract

The paper studies quasilinear elliptic problems in the Sobolev spaces W 1,p (Ω), Ω⊂RN , with p = N, that is, the case of Pohozhaev-Trudinger-Moser inequality. Similarly to the case p < N where the loss of compactness in W 1,p (RN) occurs due to dilation operators u→t(N-p)/Pu(tx), t > 0, and can be accounted for in decompositions of the type of Struwe's "global compactness" and its later refinements, this paper presents a previously unknown group of isometric operators that leads to loss of compactness in W01,N over a ball in RN. We give a one-parameter scale of Hardy-Sobolev functionals, a "p = N"-counterpart of the Holder interpolation scale, for p > N, between the Hardy functional ∫|u|p/|x|pdx and the Sobolev functional ∫|u|pN/(N-mp)dx. Like in the case p < N, these functionals are invariant with respect to the dilation operators above, and the respective concentration-compactness argument yields existence of minimizers for W 1,N -norms under Hardy-Sobolev constraints.

Item Type:Article
Source:Copyright of this article belongs to Birkhauser-Verlag.
Keywords:Trudinger-moser Inequality; Elliptic Problems in Two Dimensions; Concentration Compactness; Global Compactness; Asymptotic Orthogonality; Weak Convergence; Palais-smale Sequences
ID Code:12083
Deposited On:10 Nov 2010 04:42
Last Modified:09 May 2011 11:22

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