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The paper studies quasilinear elliptic problems in the Sobolev spaces W1,p(Ω), $${\Omega\subset{\mathbb R}^N}$$ , with p = N, that is, the case of Pohozhaev–Trudinger–Moser inequality. Similarly to the case p < N where the loss of compactness in $${W^{1,p}({\mathbb R}^N)}$$ occurs due to dilation operators $${u {\mapsto}t^{(N-p)/p}u(tx)}$$ , 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 $${{\mathbb R}^N}$$ . We give a one-parameter scale of Hardy–Sobolev functionals, a “p = N”-counterpart of the Hölder interpolation scale, for p > N, between the Hardy functional $${\int \frac{|u|^p}{|x|^p}\,{\rm d}x}$$ and the Sobolev functional $${\int |u|^{pN/(N-mp)} \,{\rm d}x}$$ . 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 W1,N-norms under Hardy–Sobolev constraints.

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