Cocompactness and minimizers for inequalities of Hardy-Sobolev type involving N-Laplacian
2010 (English)In: NoDEA. Nonlinear differential equations and applications (Printed ed.), ISSN 1021-9722, E-ISSN 1420-9004, Vol. 17, no 4, 467-477 p.Article in journal (Refereed) Published
The paper studies quasilinear elliptic problems in the Sobolev spaces W-1,W-p(Omega), Omega subset of R-N, 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,W-p(R-N) occurs due to dilation operators u bar right arrow t((N-p)/p)u(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 W-0(1,N) over a ball in R-N. 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 integral vertical bar u vertical bar(p)/vertical bar x vertical bar(p) dx and the Sobolev functional integral vertical bar u vertical bar(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,W-N-norms under Hardy-Sobolev constraints.
Place, publisher, year, edition, pages
2010. Vol. 17, no 4, 467-477 p.
Trudinger-Moser inequality, Elliptic problems in two dimensions, Concentration compactness, Global compactness, Asymptotic orthogonality, Weak convergence, Palais-Smale sequences
IdentifiersURN: urn:nbn:se:uu:diva-135597DOI: 10.1007/s00030-010-0063-4ISI: 000280641800005OAI: oai:DiVA.org:uu-135597DiVA: diva2:375243