  Sufficient and necessary conditions for the generalized Gagliardo-Nirenberg
(GN) inequality in Besov spaces and Triebel-Lizorkin spaces are obtained.
Applying the GN inequality, we show that the finite-time blowup solutions have
concentration phenomena in critical Lebesgue space L^3. Moreover, we consider
the minimizer for a class of variational problem by applying the fractional GN
inequality.
