On the Caffarelli-Kohn-Nirenberg inequalities: Sharp constants, existence (and nonexistence), and symmetry of extremal functions

Abstract

Consider the following inequalities due to Caffarelli, Kohn, and Nirenberg [6]:. 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.

A global compactness result for singular elliptic problems involving critical Sobolev exponent

“…In the following, let [4], the solutions of (1.2) are in one-to-one correspondence to solutions of the following type of problem:…”

On the Caffarelli–Kohn–Nirenberg inequalities

“…The idea is to use a conformal transformation to convert the problem to an equivalent one defined in a domain on a cyliner C ¼ R Â S NÀ1 : This idea has been used in [8] to study the symmetry property of extremal functions for the Caffarelli-Kohn-Nirenberg inequalities (1). More precisely, to a function uAC N 0 ðO\f0gÞ we associate vAC N 0 ð * OÞ by the transformation…”

“…In [8], it was proved that when O ¼ R N ; the above transformation defines a Hilbert space isomorphism between D 1;2 a ðR N Þ and H 1 ðCÞ whose norm is given by jjvjj 2…”

“…Our approach is quite different from that in [5,14], in some sense simpler and easier to be adapted for the weighted versions. Following the idea used in [8], we convert the problem from R N to one defined on a cylinder C ¼ R Â S NÀ1 : From there an inequality similar to the classical one-dimensional Hardy inequality on ð0; NÞ is used to tackle the technical part of the proof. We also note that while the sharpness of Theorems A and B is open-ended (for qo 2N NÀ2 and qo2; respectively), the sharpness in Theorem 1 is close-ended in the sense ln R with jgðxÞj-N as jxj-0 when 0AO (by gðxÞðln R jxj Þ À1 with jgðxÞj-N as jxj-N when B C r ð0ÞCO).…”

Sobolev inequalities for the Hardy–Schrödinger operator: extremals and critical dimensions

“…Also, if 0 < s < 2, the constant μ 0,s (R n ) is again explicit, and the extremals are also known (see Ghoussoub and Yuan [33], Lieb [44], Catrina and Wang [15]). More precisely, μ 0,s (R n ) = (n − 2)(n − s) ω n−1 2 − s · 2 ( n−s 2−s ) ( 2n−2s 2−s )…”