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

Abstract

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)/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 1,N 0 over a ball in 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 |u| p |x| p dx 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 HardySobolev constraints. Mathematics Subject Classification (2000). Primary 35J20, 35J35, 35J60; Secondary 46E35, 47J30, 58E05.

On a version of Trudinger–Moser inequality with Möbius shift invariance

“…In this paragraph we summarize results of [3]. Let H |x| 2 dx and the critical nonlinearity |u| 2 * dx.…”

A subset of Caffarelli–Kohn–Nirenberg inequalities in the hyperbolic space $${{\mathbb {H}}}^N$$ H N

“…The case N = 2 of the present paper, once one identifies the space H 1 0 (B) of the open unit disk B ⊂ R 2 as the Sobolev space of the Poincaré disk model of H 2 , has been already studied in the paper [3]. The scale-invariant inequalities in [3] provide bounds for appropriate weighted L p -norms of a function, or its spherical decreasing rearrangement, by the L N -norm of its gradient on the N -dimensional ball.…”

“…Similarly to the original CKN inequalities, which are invariant (up to a normalization factor) with respect to linear scalings {u(x) → u(t x)} t>0 , their counterparts in [3], in restriction to radial functions, are invariant up to normalization with respect to nonlinear scalings {u(r ) → u(r s )} s>0 . In this paper, we show that this transformation is a particular case of the scaling transformation r → G −1 (λG(r )) where G(r ) is the radial fundamental solution (which can be obviously taken here up to an arbitrary scalar multiple) of the Poisson equation for the hyperbolic Laplace-Beltrami operator.…”

“…The scale-invariant inequalities in [3] provide bounds for appropriate weighted L p -norms of a function, or its spherical decreasing rearrangement, by the L N -norm of its gradient on the N -dimensional ball. The inequalities for general values p ≥ N are derived, without losing scaling invariance, from the corresponding inequalities for p = N and p = ∞ by means of Hölder inequality.…”

“…A refined analysis of concentration compactness phenomena, in the form of profile decomposition, is given for sequences of radial functions. In Appendix, we provide cross-references between the results of this paper and the results in [3] for the case N = 2, one of them being representation of the Moser-Trudinger inequality as a two-dimensional case of the Sobolev embedding for the hyperbolic space.…”