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?



Extremal functions for Gagliardo-Nirenberg inequality
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. 


