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- W1989852008 abstract "Consider the following inequalities due to Caffarelli, Kohn and Nirenberg [3]: (∫RN|x|−bp|u|pdx)2/p≤Ca,b∫RN|x|−2a|∇u|2dx, where for N≥3: −∞<a<N−22, a≤b≤a+1, and p=2NN−2+2(b−a). We shall answer some fundamental questions concerning these inequalities such as the best embedding constants, the existence and nonexistence of extremal functions, and their qualitative properties. We also study the bound state solutions of the corresponding Euler equations and construct positive solutions having prescribed symmetry for certain parameter region. On considère les inégalités suivantes de Caffarelli, Kohn et Nirenberg [3] : (∫RN|x|−bp|u|pdx)2/p≤Ca,b∫RN|x|−2a|∇u|2dx, où N≥3: −∞<a<N−22, a≤b≤a+1 et p=2NN−2+2(b−a). Nous répondons à quelques questions fondamentales concernant ces inégalités, telles que meilleures constantes d'injection, existence ou non-existence des fonctions extrémales et leur propriétés qualitatives. On étudie aussi les solutions à état borné qui correspondent à l'équation d'Euler et on construit des solutions positives avec une symétrie prescrite dans certaines régions de l'espace des paramètres." @default.
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- W1989852008 date "2000-03-01" @default.
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- W1989852008 title "On the Caffarelli–Kohn–Nirenberg inequalities" @default.
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- W1989852008 doi "https://doi.org/10.1016/s0764-4442(00)00201-9" @default.
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