About Existence, Symmetry and Symmetry Breaking for Extremal Functions of Some Interpolation Functional Inequalities

This chapter 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 (Dolbeault et al. in Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 7(2):313–341, 2008; Dolbeault et al. in Adv. Nonlinear Stud. 9(4):713–726, 2009; Dolbeault, Esteban in Extremal functions for Caffarelli–Kohn–Nirenberg and logarithmic Hardy inequalities Calc. Var. Partial Differ. Equ., 2010; del Pino et al. in J. Funct. Anal. 259(8):2045–2072, 2010; Dolbeault et al. in Radial symmetry and symmetry breaking for some interpolation inequalities, preprint, 2010) 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).