The best constant of the Lp Sobolev-type inequality corresponding to elliptic operator in RN

Abstract

The $L^p$ Sobolev-type inequality shows that the supremum of $\left|u\left(y\right)\right|$ defined on ${\mathbf{R}}^N$ is estimated from above by constant $C$ multiples of the $L^p$ norm of $\left(-\Delta +a^2\right)u\left(x\right)$. Among such constant $C$, the smallest constant is the best constant $C_0$. If we replace $C$ by $C_0$ in the $L^p$ Sobolev-type inequality, then the equality holds for the best function $U\left(x\right)$. The aim of this paper is to find $C_0$ and $U\left(x\right)$ of the $L^p$ Sobolev-type inequality. The Green function $G\left(x-y\right)$ of partial differential equation of elliptic type $\left(-\Delta +a^2\right)u\left(x\right)=f\left(x\right)$ defined on ${\mathbf{R}}^N$ is an important factor in this paper because $C_0$ and $U\left(x\right)$ consist of the Green function.