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

Abstract

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 - 1 p - 1 {r=p\frac{q-1}{p-1}} and q = p ⁢ r - 1 p - 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.