Search results for " Liouville-type theorem"

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

2022

Abstract 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.

Pure mathematicsApplied MathematicsOperator (physics)Caffarelli–Kohn–Nirenberg inequalities Classification of solutions Liouville-type theorem Optimal constant Quasilinear anisotropic elliptic equationsMathematics::Analysis of PDEsType (model theory)Range (mathematics)Settore MAT/05 - Analisi MatematicaSymmetry breakingSymmetry (geometry)Nirenberg and Matthaei experimentLaplace operatorAnalysisMathematics
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Landis-type conjecture for the half-Laplacian

2023

In this paper, we study the Landis-type conjecture, i.e., unique continuation property from infinity, of the fractional Schrödinger equation with drift and potential terms. We show that if any solution of the equation decays at a certain exponential rate, then it must be trivial. The main ingredients of our proof are the Caffarelli-Silvestre extension and Armitage’s Liouville-type theorem. peerReviewed

Landis conjecture half-Laplacian Caarelli- Silvestre extension Liouville-type theoremosittaisdifferentiaaliyhtälötMathematics - Analysis of PDEsApplied MathematicsGeneral Mathematicsunique continuation propertyPrimary: 35A02 35B40 35R11. Secondary: 35J05 35J15FOS: MathematicsAnalysis of PDEs (math.AP)
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