Fractional Sharp Gagliardo-Nirenberg Inequalities With Weights

In this master dissertation,using the variational method,with the help of the fractional order Caffarelli-Kohn-Nirenberg inequality,we prove the following optimal weighted fractional order Gagliardo-Nirenberg inequality|||x‖k/(2σ+2)u‖L2σ+2≤Copt‖u‖L2(1-θ)‖(-△)s/2u‖L2θ,where k,σ satisfy 1)k∈(max{-2s,-N},N-2s),σ∈[(2s+k)/N,(2s+k)/(N-2s)]or 2)k ∈(-2s,0),σ∈[0,(2s+k)/(N-2s)],θ=(Nσ-k)/(s(2σ+2)).Copt is the optimal constant.This result is about the generalization of the result of s=1 in the fractional order case.As an application,we proved a class of non-homogeneous,non-homogeneous fractional order in the critical case of L2 Schrodinger equation explosion cracking at the moment of blast L2 nonexistence.