On the best constant for Gagliardo-Nirenberg interpolation inequalities

Abstract

In this paper we derive the best constant for the following Gagliardo-Nirenberg interpolation inequality \begin{eqnarray*} \|u\|_{L^{m+1}}\leq C_{q,m,p} \|u\|^{1-θ}_{L^{q+1}}\|\nabla u\|^θ_{L^p},\quad θ=\frac{pd(m-q)}{(m+1)[d(p-q-1)+p(q+1)]}, \end{eqnarray*} where parameters $q,m,p$ respectively belong to the following two ranges: (i) $p>d\ge 1$, $q\ge 0$ and $m=\infty$. That shows $L^{\infty }$-type Gagliardo-Nirenberg interpolation inequality. (ii) $p>\max \{1,\frac{2d}{d+2}\}$, $0\le q<\sigma -1$, and $q<m<\sigma$, where $\sigma$ is defined by $\sigma :=\frac{(p-1)d+p}{d-p}$ if $p<d$; $\sigma :=\infty$ if $p\ge d$. That gives $L^m$-type Gagliardo-Nirenberg interpolation inequality. The best constant $C_{q,m,p}$ is given by \begin{eqnarray*} C_{q,m,p}:=θ^{-\fracθ{p}}(1-θ)^{\fracθ{p}-\frac{1}{m+1}}M_c^{-\fracθ{d}},\quad M_c:=\int_{\mathbb{R}^d}u_{c,m}^{q+1}\,dx, \end{eqnarray*} where $u_{c,m}$ is the unique radial non-increasing solution to a generalized Lane-Emden equation. The case of equality holds when $u=A{u}_{c,m}(\lambda (x-x_0))$ for any real numbers $A>0$, $\lambda >0$ and $x_0\in {\mathbb{R}}^d$. In particular, for the case $m=+\infty$, the generalized Lane-Emden equation becomes a Thomas-Fermi type equation. For $q=0,m=\infty$ or $d=1$, $u_{c,m}$ are closed form solutions expressed in term of the incomplete Beta functions. Moreover, we show that $u_{c,m}\to u_{c,\infty }$ and $C_{q,m,p}\to C_{q,\infty ,p}$ as $m\to +\infty$ for $d=1$.

References in corpus (1)

• Qualitative properties and existence of sign changing solutions with compact support for an equation with a p-Laplace operator