On the Caffarelli-Kohn-Nirenberg inequalities: Sharp constants, existence (and nonexistence), and symmetry of extremal functions †
On the Caffarelli-Kohn-Nirenberg inequalities: Sharp constants, existence (and nonexistence), and symmetry of extremal functions †
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DOI：
10.1002/1097-0312(200102)54:2
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发表时间：
2001-02
影响因子：
3
通讯作者：
Florin Catrina;Zhi-Qiang Wang
中科院分区：
文献类型：
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作者：
Florin Catrina;Zhi-Qiang Wang
考虑以下由Caffarelli、Kohn和Nirenberg [6]提出的不等式，其中对于N ≥ 3，−∞ < a <（N − 2）/2，a ≤ B ≤ a + 1，且p = 2N/（N − 2 + 2（B − a））。我们将回答有关这些不等式的一些基本问题，如最佳嵌入常数，极值函数的存在性和不存在性，以及它们的定性性质。虽然a ≥ 0的情况已经被广泛研究，并且已知完整的解决方案，但对于a < 0的情况知之甚少。我们对a < 0情形的结果揭示了一些新的现象，这些现象与a ≥ 0情形的结果形成了鲜明的对比.给出了N = 1和N = 2时的结果。John Wiley & Sons，Inc.
Consider the following inequalities due to Caffarelli, Kohn, and Nirenberg [6] where, for N ≥ 3, −∞ < a < (N − 2)/2, a ≤ b ≤ a + 1, and p = 2N/(N − 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. While the case a ≥ 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 ≥ 0. Results for N = 1 and N = 2 are also given. © 2001 John Wiley & Sons, Inc.