Blow-up solutions for the inhomogeneous Schrödinger equation with L2 supercritical nonlinearity

This paper studies blow-up solutions for the inhomogeneous Schrödinger equation with L2 supercritical nonlinearity. In terms of Strauss' arguments in Strauss (1977)  [22], we find a new compactness lemma for radial symmetric functions. Thus, we use it to derive the best constants of two generalized Gagliardo-Nirenberg type inequalities. Moreover, we obtain a more precisely sharp criteria of blow-up and global existence, and derive the weak concentration phenomenon of blow-up solutions by the variational methods.