Extremal functions for Gagliardo-Nirenberg inequality Recently I read about the Gagliardo-Nirenberg inequality. And I would like to ask about the attainability and the maximizers of the GN inequality:
$(∫|u|^{r}dx)^{\frac{1}{r}} \leq GN(N,p,q,r)(∫|∇u|^{p}dx)^{\frac{a}{p}}(∫|u|^{q}dx)^{\frac{1-a}{q}}$. 
Can the best constant GN(N,p,q,r) be achieved in some cases? Does anyone know good references about this subject?
 A: There is a $1$--parameter family of inequalities where the sharp constants and corresponding extremal functions are known. I believe this was first established by Del Pino and Dolbeault. Cordero, Nazaret, and Villani gave a beautiful optimal transportation proof. See their paper for the relevant references. The family includes the sharp Sobolev and sharp log-Sobolev inequalities. 
ADDED: Here are the exactly references:
Del Pino, Manuel(RCH-UCSP-EM); Dolbeault, Jean(F-PARIS9-A)
Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions.
J. Math. Pures Appl. (9) 81 (2002), no. 9, 847–875. 
Cordero-Erausquin, D.(F-MARN-AMA); Nazaret, B.(F-ENSLY-PM); Villani, C.(F-ENSLY-PM)
A mass-transportation approach to sharp Sobolev and Gagliardo-Nirenberg inequalities.
Adv. Math. 182 (2004), no. 2, 307–332. 
A: As a complement to Deane Yang's answer: arguably a special case of major interest is $p = q = 2$. For this case, existence of Gagliardo-Nirenberg maximizers was proved in $H^1(\mathbb{R}^n)$ by Michael Weinstein in this paper: http://www.ams.org/mathscinet-getitem?mr=691044.
The basic argument was, using spatial scaling in $\mathbb{R}^n$, consider a maximizing sequence $\varphi_n$ such that $\Vert \varphi_n\Vert_{L^2} = 1 = \Vert \nabla \varphi_n\Vert_{L^2} $. Then, since $\varphi_n$ is weakly $H^1$-bounded, the weak limit $u$ is shown to be (using Sobolev embedding) the maximizer. Note that this approach does not work for the hyperbolic space, for example, where  spatial dilation is not available. The preprint http://arxiv.org/abs/1406.4931 shows that the GN maximizer ($p = q = 2$) does not exist in $H^1(\mathbb{H}^n)$.
