Symmetry for positive critical points of Caffarelli–Kohn–Nirenberg inequalities

• Published in: Nonlinear analysis Vol. 216; p. 112683
• Main Authors: Ciraolo, Giulio, Corso, Rosario
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.03.2022
• Subjects: Caffarelli–Kohn–Nirenberg inequalities; Classification of solutions; Liouville-type theorem; Optimal constant; Quasilinear anisotropic elliptic equations; Caffarelli–Kohn–Nirenberg inequalities; Liouville-type theorem; Quasilinear anisotropic elliptic equations; 35J92; 53C21; 35B53; Optimal constant; Classification of solutions; 35B09
• ISSN: 0362-546X; 1873-5215
• DOI: 10.1016/j.na.2021.112683

We consider positive critical points of Caffarelli–Kohn–Nirenberg inequalities and prove a Liouville type result which allows us to give a complete classification of the solutions in a certain range of parameters, providing a symmetry result for positive solutions. The governing operator is a weighted p-Laplace operator, which we consider for a general p∈(1,d). For p=2, the symmetry breaking region for extremals of Caffarelli–Kohn–Nirenberg inequalities was completely characterized in Dolbeault et al. (2016). Our results extend this result to a general p and are optimal in some cases.