## Kyoto Journal of Mathematics

### On the Caffarelli–Kohn–Nirenberg-type inequalities involving critical and supercritical weights

#### Abstract

The main purpose of this article is to establish the Caffarelli–Kohn– Nirenberg-type (CKN-type) inequalities for all $\alpha \in \mathbf {R}$ and to study the related matters systematically. Roughly speaking, we discuss the characterizations of the CKN-type inequalities for all $\alpha \in \mathbf {R}$ as the variational problems, the existence and nonexistence of the extremal solutions to these variational problems in proper spaces, and the exact values and the asymptotic behaviors of the best constants in both the noncritical case and the critical case.

In the study of the CKN-type inequalities, the presence of weight functions on both sides prevents us from employing effectively the so-called spherically symmetric rearrangement. Further the invariance of $\mathbf {R}^{n}$ by the group of dilatations creates some possible loss of compactness. As a result we see that the existence of extremals, the values of best constants, and their asymptotic behaviors essentially depend upon the relations among parameters in the inequality.

#### Article information

Source
Kyoto J. Math., Volume 52, Number 4 (2012), 661-742.

Dates
First available in Project Euclid: 15 November 2012

Permanent link to this document
https://projecteuclid.org/euclid.kjm/1352987531

Digital Object Identifier
doi:10.1215/21562261-1728839

Mathematical Reviews number (MathSciNet)
MR2872207

Zentralblatt MATH identifier
1267.46051

Subjects
Primary: 35J70: Degenerate elliptic equations
Secondary: 35J60: Nonlinear elliptic equations

#### Citation

Horiuchi, Toshio; Kumlin, Peter. On the Caffarelli–Kohn–Nirenberg-type inequalities involving critical and supercritical weights. Kyoto J. Math. 52 (2012), no. 4, 661--742. doi:10.1215/21562261-1728839. https://projecteuclid.org/euclid.kjm/1352987531

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