The best constant for 𝐿^{∞}-type Gagliardo-Nirenberg inequalities

Abstract

In this paper we derive the best constant for the followingL∞L^{\infty }-type Gagliardo-Nirenberg interpolation inequality‖u‖L∞≤Cq,∞,p‖u‖Lq+11−θ‖∇u‖Lpθ,θ=pddp+(p−d)(q+1),\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 parametersqqandppsatisfy the conditionsp>d≥1p>d\geq 1,q≥0q\geq 0. The best constantCq,∞,pC_{q,\infty ,p}is given byCq,∞,p=θ−θp(1−θ)θpMc−θd,Mc≔∫Rduc,∞q+1dx,\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*}whereuc,∞u_{c,\infty }is the unique radial non-increasing solution to a generalized Lane-Emden equation. The case of equality holds whenu=Auc,∞(λ(x−x0))u=Au_{c,\infty }(\lambda (x-x_0))for any real numbersAA,λ>0\lambda >0andx0∈Rdx_{0}\in \mathbb {R}^d. In fact, the generalized Lane-Emden equation inRd\mathbb {R}^dcontains a delta function as a source and it is a Thomas-Fermi type equation. Forq=0q=0ord=1d=1,uc,∞u_{c,\infty }have closed form solutions expressed in terms of the incomplete Beta functions. Moreover, we show thatuc,m→uc,∞u_{c,m}\to u_{c,\infty }andCq,m,p→Cq,∞,pC_{q,m,p}\to C_{q,\infty ,p}asm→+∞m\to +\inftyford=1d=1, whereuc,mu_{c,m}andCq,m,pC_{q,m,p}are the function achieving equality and the best constant ofLmL^m-type Gagliardo-Nirenberg interpolation inequality, respectively.