Existence of Extremal Functions for Higher-Order Caffarelli–Kohn–Nirenberg Inequalities

Abstract

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 ( ∫ ℝ N | D j u | r d ⁢ x | x | s ) 1 r ≤ C ( ∫ ℝ N | D m u | p d ⁢ x | x | μ ) a p ( ∫ ℝ N | u | q d ⁢ x | x | σ ) 1 - a q , \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 , μ , σ , s , m , j ) {C=C(p,q,r,\mu,\sigma,s,m,j)} and p , q , r , μ , σ , s , m , j {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 H ˙ μ m , p ⁢ ( Ω ) ∩ L σ q ⁢ ( Ω ) ↪ H ˙ s j , r ⁢ ( Ω ) {\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.