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?