A tool for symmetry breaking and multiplicity in some nonlocal problems

Abstract

We prove some basic inequalities relating the Gagliardo-Nirenberg seminorms of a symmetric function u on R n and of its perturbation uϕ µ , where ϕ µ is a suitably chosen eigenfunction of the Laplace-Beltrami operator on the sphere S n−1 , thus providing a technical but rather powerful tool to detect symmetry breaking and multiplicity phenomena in variational equations driven by the fractional Laplace operator. A concrete application to a problem related to the fractional Caffarelli-Kohn-Nirenberg inequality is given.

The $s$-polyharmonic extension problem and higher-order fractional Laplacians

“…by Herbst [8]. The next Theorem generalizes [12, Lemma 2.1], see also [13, Theorem 1], to higher orders, and gives a positive answer to a question raised in [12,Remark 2.2] for n = 1, s ∈ (0, 1 2 ).…”

“…6. The proof of Theorem 1.3 differs from that in [12,13]; it relies on the characterization of Muckhenoupt weights via the Hardy-Littlewood maximal operator. We believe that it can be further generalized in order to consider additional parameters, in the spirit of [13]; nevertheless, this is beyond the aim of the present work.…”

Complete classification and nondegeneracy of minimizers for the fractional Hardy-Sobolev inequality, and applications

“…By (5.1), u λ h is a bounded minimizing sequence for S q , so we can suppose that u λ h converges weakly in D s . Arguing as in the proof of [16,Lemma 4.2], we can rescale u λ h so that its weak limit u is non-zero, hence u is a (nonnegative) solution of (P 0 ) and u λ h → u strongly in D s . Thanks to the uniqueness result in Theorem 1.1, we see that there exists τ > 0 and a sequence…”

“…is radial, provided that −H s < λ ≤ 0 (existence has been proved in [16]). On the other hand, symmetry breaking occurs: if λ > 0 is large, then no extremal for S λ q is radially symmetric (see [16,Theorem 1.1]).…”

“…(see [16]) which is a weak solution to (−∆) s u = |x| −bq u q−1 . Since (−∆) s z 1 ≥ 0 in the sense of distributions, then the strong maximum principle (see [19,Section 2] and [15,Corollary 4.2]), ensures that z 1 is lower semicontinuous and positive on R n .…”