About existence, symmetry and symmetry breaking for extremal functions of some interpolation functional inequalities

2010/11/24 by Dolbeault, Jean, Esteban, Maria J. · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1011.5465

Abstract

This article is devoted to a review of some recent results on existence, symmetry and symmetry breaking of optimal functions for Caffarelli-Kohn-Nirenberg (CKN) and weighted logarithmic Hardy (WLH) inequalities. These results have been obtained in a series of papers in collaboration with M. del Pino, S. Filippas, M. Loss, G. Tarantello and A. Tertikas. Here we put the highlights on a symmetry breaking result: extremals of some inequalities are not radially symmetric in regions where the symmetric extremals are linearly stable. Special attention is paid to the study of the critical cases for (CKN) and (WLH).

Cited by

• Discrete Caffarelli-Kohn-Nirenberg inequalities and ground state solutions to nonlinear elliptic equations

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