Abstract
Though there has been an extensive study on first-order Caffarelli–Kohn–Nirenberg inequalities, not much is known for the existence of extremal functions for higher-order ones.
The higher-order derivative of the Caffarelli–Kohn–Nirenberg inequality established by Lin [14] states
\bigg{(}\int_{\mathbb{R}^{N}}\lvert D^{j}u|^{r}\frac{dx}{|x|^{s}}\bigg{)}^{%
\frac{1}{r}}\leq C\bigg{(}\int_{\mathbb{R}^{N}}\lvert D^{m}u|^{p}\frac{dx}{|x|%
^{\mu}}\bigg{)}^{\frac{a}{p}}\bigg{(}\int_{\mathbb{R}^{N}}\lvert u|^{q}\frac{%
dx}{|x|^{\sigma}}\bigg{)}^{\frac{1-a}{q}},
where
{C=C(p,q,r,\mu,\sigma,s,m,j)}
and
{p,q,r,\mu,\sigma,s,m,j}
are parameters satisfying some balanced conditions. The main purpose of this paper is to establish the existence of extremal functions for a family of this higher-order derivatives of Caffarelli–Kohn–Nirenberg inequalities under numerous circumstances of parameters.
Moreover, we study the compactness of the weighted Sobolev space for higher-order derivatives and prove that
{\dot{H}^{m,p}_{\mu}(\Omega)\cap L^{q}_{\sigma}(\Omega)\hookrightarrow\dot{H}^%
{j,r}_{s}(\Omega)}
is a compact embedding within some range of the parameters.
ON THE SOLVABILITY OF HIGHER-ORDER OPERATOR-DIFFERENTIAL EQUATIONS IN A WEIGHTED SOBOLEV SPACE
