Sharp constants and optimizers for a class of the Caffarelli-Kohn-Nirenberg inequalities

In this paper, we will use a suitable tranform to investigate the sharp constants and optimizers for the following Caffarelli-Kohn-Nirenberg inequalities for a wide range of parameters $(r,p,q,s,\mu,\sigma)$ and $0\leq a\leq1$: \begin{equation} \left({\displaystyle\int} \left\vert u\right\vert ^{r}\frac{dx}{\left\vert x\right\vert ^{s}}\right)^{1/r}\leq C\left( {\displaystyle\int} \left\vert \nabla u\right\vert ^{p}\frac{dx}{\left\vert x\right\vert ^{\mu}% }\right) ^{a/p}\left({\displaystyle\int} \left\vert u\right\vert ^{q}\frac{dx}{\left\vert x\right\vert ^{\sigma}% }\right) ^{\left( 1-a\right) /q}. \end{equation} We are able to compute the best constants and the explicit forms of the extremal functions in numerous cases. When $0<a<1$, we can deduce the existence and symmetry of optimizers for a wide range of parameters. Moreover, in the particular classes $r=p\frac{q-1}{p-1}$ and $q=p\frac{r-1}{p-1}$, the forms of maximizers will also be provided in the spirit of Del Pino and Dolbeault ([12], 13]). In the case $a=1$, that is the Caffarelli-Kohn-Nirenberg inequality without the interpolation term, we will provide the exact maximizers for all the range of $\mu\geq0$. The Caffarelli-Kohn-Nirenberg inequalities with arbitrary norms on the Euclidean spaces will also be considered in the spirit of Cordero-Erausquin, Nazaret and Villani [10].