Sharp Constants and Optimizers for a Class of Caffarelli–Kohn–Nirenberg Inequalities

By Nguyen Huong Lam, G. Lu. 2015

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Abstract In this paper, we use a suitable transform of quasi-conformal mapping type to investigate the sharp constants and optimizers for the following Caffarelli–Kohn–Nirenberg inequalities for a large class of parameters (r,p,q,s,μ,σ){(r,p,q,s,\mu,\sigma)} and 0≤a≤1{0\leq a\leq 1}: ( ∫ | u | r d ⁢ x | x | s ) 1 r ≤ C ( ∫ | ∇ u | p d ⁢ x | x | μ ) a p ( ∫ | u | q d ⁢ x | x | σ ) 1 - a q . \bigg{(}\int\lvert u|^{r}\frac{dx}{|x|^{s}}\bigg{)}^{\frac{1}{r}}\leq C\bigg{(% }\int\lvert\nabla u|^{p}\frac{dx}{\lvert x|^{\mu}}\bigg{)}^{\frac{a}{p}}\bigg{% (}\int\lvert u|^{q}\frac{dx}{\lvert x|^{\sigma}}\bigg{)}^{\frac{1-a}{q}}. We compute the best constants and the explicit forms of the extremal functions in numerous cases. When 0<a<1{0<a<1}, we can deduce the existence and symmetry of optimizers for a wide range of parameters. Moreover, in the particular cases r=p⁢q-1p-1{r=p\frac{q-1}{p-1}} and q=p⁢r-1p-1{q=p\frac{r-1}{p-1}}, the forms of maximizers will also be provided in the spirit of Del Pino and Dolbeault [14, 15]. In the case a=1{a=1}, that is, the Caffarelli–Kohn–Nirenberg inequality without the interpolation term, we will provide the exact maximizers for all the range of μ≥0{\mu\geq 0}. The Caffarelli–Kohn–Nirenberg inequalities with arbitrary norms on Euclidean spaces will also be considered in the spirit of Cordero-Erausquin, Nazaret and Villani [13]. Due to the absence of the classical Polyá–Szegö inequality in the weighted case, we establish a symmetrization inequality with power weights which is of independent interest.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Geometry and Topology

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