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

Abstract

The paper studies quasilinear elliptic problems in the Sobolev spaces W ^1,p (Ω), Ω⊂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,p (R^N) 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 W₀^1,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 ∫|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.