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arXiv · 1907.09182

A tool for symmetry breaking and multiplicity in some nonlocal problems

Roberta Musina·Alexander I. Nazarov
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Abstract

We prove some basic inequalities relating the Gagliardo-Nirenberg seminorms of a symmetric function $u$ on $\mathbb R^n$ and of its perturbation $u\varphi_\mu$, where $\varphi_\mu$ is a suitably chosen eigenfunction of the Laplace-Beltrami operator on the sphere $\mathbb S^{n-1}$, 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.

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Roberta Musina, Alexander I. Nazarov. 2019-07-22. A tool for symmetry breaking and multiplicity in some nonlocal problems. https://doi.org/10.1002/mma.6220

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