The main purpose of this article is to establish the Caffarelli–Kohn– Nirenberg-type (CKN-type) inequalities for all and to study the related matters systematically. Roughly speaking, we discuss the characterizations of the CKN-type inequalities for all 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 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.
"On the Caffarelli–Kohn–Nirenberg-type inequalities involving critical and supercritical weights." Kyoto J. Math. 52 (4) 661 - 742, Winter 2012. https://doi.org/10.1215/21562261-1728839