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CAFFARELLI-KOHN-NIRENBERG INEQUALITIES 233 equation corresponding to (1.8). The two problems will be shown to be equivalent, and we shall mainly work on the transformed one on R SN−1. The advantage in working on the latter is that the equation is an autonomous one and is deﬁned in H1.R SN−1/.

http://www.math.usu.edu/~wang/cpam.pdfWe consider positive solutions of -div(|x|-2a u) = |x|-bpup-1, u ≥ 0 in RN, where for N ≥ 2: a < (N - 2)/2, a < b < a + 1, and p = 2N/(N - 2(1 + a - b)). Ground state solutions are the extremal functions of the Caffarelli-Kohn-Nirenberg inequalities [6].

https://www.researchgate.net/publication/2452776_On_the_Caffarelli...On the Caffarelli-*Kohn*-Nirenberg *inequalities*: Sharp constants, existence (and nonexistence), and symmetry of extremal functions †

http://onlinelibrary.wiley.com › … › Vol 54 Issue 2 › AbstractC. R. Acad. Sci. Paris, t. 330, Série I, p. 437–442, 2000 Équations aux dérivées partielles/Partial Differential Equations On the Caffarelli–*Kohn* ...

http://math.usu.edu/~wang/CRAS-2000.pdfIn the setting of a Lie group of polynomial volume growth, we derive *inequalities* of Caffarelli-*Kohn*-Nirenberg type, where the weights involved are powers ...

https://www.researchgate.net/publication/313797480_The Caffarelli–Kohn–Nirenberg *inequalities* state that there is a constant C > 0 such that

https://link.springer.com/article/10.1007/s00205-009-0269-yWe investigate Caffarelli–Kohn–Nirenberg-type *inequalities* for the weighted biharmonic operator in cones, both under Navier and Dirichlet boundary ...

https://link.springer.com/article/10.1007/s00032-011-0167-2inequalities (i.e., 0 < a < 1 and at least one of s,µ,θ is nonzero), see the review paper by Dolbeault and Esteban [15]. Compared with the special cases of the Gagliardo-Nirenberg inequalities without the interpolation term (i.e., a = 1), dealing with such CKN inequalities encounters considerably more diﬃculty.

https://arxiv.org/pdf/1510.01224.pdfAnother important inequality is the so called Nash inequality stating that for any f ∈ C∞ 0 (IRn) we have Z IRn f2dv 1+2/n ≤ C n IRn |∇f|2dv Z IRn |f|dv 4/n (1.8) for a constant C n depending only on n. This inequality is a particular case of the Gagliardo-Nirenberg inequalities for which numerous applications have been found.

http://intlpress.com/site/pub/files/_fulltext/journals/mrl/2007/0014/0005/...Consider the following *inequalities* due to Caffarelli, *Kohn* and Nirenberg ... We shall answer some fundamental questions concerning these *inequalities* ...

https://www.sciencedirect.com/science/article/pii/S0764444200002019- On Hardy and Caffarelli-Kohn-Nirenberg inequalities
- A subset of Caffarelli–Kohn–Nirenberg inequalities in the hyperbolic
- Some improved Caffarelli-Kohn-Nirenberg inequalities
- Extended Caffarelli-Kohn-Nirenberg inequalities
- Sharp Constants and Optimizers for a Class of Caffarelli–Kohn
- Some improvements for a class of the Caffarelli-Kohn-Nirenberg
- Norm constants in cases ofthe Caffarelli–Kohn–Nirenberg inequality
- Weighted Caffarelli-Kohn-Nirenberg type inequalities related
- Extended Caffarelli–Kohn–Nirenberg inequalities and superweights
- A Caffarelli-Kohn-Nirenberg type inequality with variable exponent