Recently I read about the Gagliardo-Nirenberg inequality. And I would like to ask about the attainability and the maximizers of the GN inequality:
$GN(N,p,q,r)=\sup\frac{∫|u|^{r}dx}{(∫|∇u|^{p}dx)^{a}(∫|u|^{q}dx)^{1-a})}$. 
Can the supremum be achieved in some cases? Does anyone know good references about this subject?