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On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions. (English) Zbl 1072.35506
Summary: Consider the inequalities due to Caffarelli, Kohn, and Nirenberg: \[ \Big(\int_{\mathbb R^N}|x|^{-bp}|u|^p\,dx\Big)^{2/p}\leq C_{a,b}\int_{\mathbb R^N}|x|^{-2a}|\nabla u|^2\,dx, \] where for \(N\geq 3\), \(-\infty<a<(N-2)/2\), \(a\leq b\leq a+1\), and \(p=2N/(N-2+2(b-a))\). We answer some fundamental questions concerning these inequalities, such as the best embedding constants, the existence and nonexistence of extremal functions, and their qualitative properties. While the case \(a\geq 0\) has been studied extensively and a complete solution is known, little has been known for the case \(a<0\). Our results for the case \(a<0\) reveal some new phenomena which are in striking contrast with those for the case \(a\geq 0\). Results for \(N=1\) and \(N=2\) are also given.

MSC:
35B33 Critical exponents in context of PDEs
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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