The best constant for $L^{\infty }$-type Gagliardo-Nirenberg inequalities

In this paper we derive the best constant for the following $L^{\infty }$-type Gagliardo-Nirenberg interpolation inequality \begin{equation*} \|u\|_{L^{\infty }}\leq C_{q,\infty ,p} \|u\|^{1-\theta }_{L^{q+1}}\|\nabla u\|^{\theta }_{L^p},\quad \theta =\frac {pd}{dp+(p-d)(q+1)}, \end{equation*} where parameters $q$ and $p$ satisfy the conditions $p>d\geq 1$, $q\geq 0$. The best constant $C_{q,\infty ,p}$ is given by \begin{equation*} C_{q,\infty ,p}=\theta ^{-\frac {\theta }{p}}(1-\theta )^{\frac {\theta }{p}}M_c^{-\frac {\theta }{d}},\quad M_c≔\int _{\mathbb {R}^d}u_{c,\infty }^{q+1} dx, \end{equation*} where $u_{c,\infty }$ is the unique radial non-increasing solution to a generalized Lane-Emden equation. The case of equality holds when $u=Au_{c,\infty }(\lambda (x-x_0))$ for any real numbers $A$, $\lambda >0$ and $x_{0}\in \mathbb {R}^d$. In fact, the generalized Lane-Emden equation in $\mathbb {R}^d$ contains a delta function as a source and it is a Thomas-Fermi type equation. For $q=0$ or $d=1$, $u_{c,\infty }$ have closed form solutions expressed in terms 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 $d=1$, where $u_{c,m}$ and $C_{q,m,p}$ are the function achieving equality and the best constant of $L^m$-type Gagliardo-Nirenberg interpolation inequality, respectively.