# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] Aubin, J Differential Geometry 11 pp 573– (1976) · Zbl 0371.46011 [2] Berestycki, J Differential Equations 134 pp 1– (1997) · Zbl 0870.34032 [3] Brezis, Proc Amer Math Soc 88 pp 486– (1983) · Zbl 0526.46037 [4] Brezis, Comm Pure Appl Math 36 pp 437– (1983) · Zbl 0541.35029 [5] Caffarelli, Comm Pure Appl Math 42 pp 271– (1989) · Zbl 0702.35085 [6] Caffarelli, Compositio Math 53 pp 259– (1984) [7] Caldiroli, Cal Var Partial Differential Equations 8 pp 365– (1999) · Zbl 0929.35045 [8] Catrina, C R Acad Sci Paris Sér I Math 330 pp 437– (2000) · Zbl 0954.35050 [9] Catrina, Ann Inst H Poincaré Anal Non Linéaire [10] Chen, Duke Math J 63 pp 615– (1991) · Zbl 0768.35025 [11] Chou, J London Math Soc (2) 48 pp 137– (1993) · Zbl 0739.26013 [12] ; Mathematical analysis and numerical methods for science and technology. Vol. 1. Physical origins and classical methods. Spinger, Berlin, 1985. [13] A review of Hardy inequalities. Preprint. · Zbl 0936.35121 [14] ; ; Symmetry of positive solutions of nonlinear elliptic equations in ?N. Mathematical analysis and applications, Part A, 369-402. Advances in Mathematics Supplemetary Studies, 7a. Academic Press, New York-London, 1981. [15] ; ; Inequalities. Second edition. Cambridge, University Press, 1952. [16] Horiuchi, J Inequal Appl 1 pp 275– (1997) [17] Korevaar, Invent Math 135 pp 233– (1999) · Zbl 0958.53032 [18] Lieb, Ann of Math (2) 118 pp 349– (1983) · Zbl 0527.42011 [19] Lin, Comm Partial Differential Equations 11 pp 1515– (1986) · Zbl 0635.46032 [20] Lin, Trans Amer Math Soc 332 pp 775– (1992) [21] Lions, Ann Inst H Poincaré Anal Non Linéaire 1 pp 109– (1984) · Zbl 0541.49009 [22] Lions, Rev Mat Iberoamericana 1 pp 145– (1985) · Zbl 0704.49005 [23] Talenti, Ann Mat Pura Appl (4) 110 pp 353– (1976) · Zbl 0353.46018 [24] Wang, J Differential Equations 159 pp 102– (1999) · Zbl 1005.35083 [25] Wang, J Differential Equations 161 pp 307– (2000) · Zbl 0954.35074 [26] Minimax theorems. Progress in Nonlinear Differential Equations and Their Applications, 24. Birkhäuser, Boston, 1996. [27] A decomposition lemma and critical minimization problems. Preprint. · Zbl 1194.35122
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.