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تسجيل مستخدم جديدSymmetry for positive critical points of Caffarelli-Kohn-Nirenberg inequalities
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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 $p in (1,d)$. For $p=2$, the symmetry breaking region for extremals of Caffarelli-Kohn-Nirenberg inequalities was completely characterized in [J. Dolbeault, M. Esteban, M. Loss, Invent. Math. 44 (2016)]. Our results extend this result to a general $p$ and are optimal in some cases.
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In this paper, we consider the Caffarelli-Kohn-Nirenberg (CKN) inequality: begin{eqnarray*} bigg(int_{{mathbb R}^N}|x|^{-b(p+1)}|u|^{p+1}dxbigg)^{frac{2}{p+1}}leq C_{a,b,N}int_{{mathbb R}^N}|x|^{-2a}| abla u|^2dx end{eqnarray*} where $Ngeq3$, $a<frac
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