On the best constant for Gagliardo-Nirenberg interpolation inequalities

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-\theta}_{L^{q+1}}\|\nabla u\|^{\theta}_{L^p},\quad \theta=\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) $p>d\geq 1$, $q\geq0$ and $m=\infty$. That shows $L^{\infty}$-type Gagliardo-Nirenberg interpolation inequality. (ii) $p>\max\{1,\frac{2d}{d+2}\}$, $0\leq q<\sigma-1$, and $q<m<\sigma$, where $\sigma$ is defined by $\sigma:= \frac{(p-1)d+p }{d-p}$ if $p<d$; $\sigma:=\infty$ if $p\geq d$. 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}:=\theta^{-\frac{\theta}{p}}(1-\theta)^{\frac{\theta}{p}-\frac{1}{m+1}}M_c^{-\frac{\theta}{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 $u=Au_{c,m}(\lambda(x-x_0))$ 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 $d=1$.