Biharmonic nonlinear Schrödinger equation and the profile decomposition

This paper is concerned with the Cauchy problem for the biharmonic nonlinear Schrödinger equation with L2L2-super-critical nonlinearity. By establishing the profile decomposition of bounded sequences in H2(RN)H2(RN), the best constant of a Gagliardo–Nirenberg inequality is obtained. Moreover, a sufficient condition for the global existence of the solution to the biharmonic nonlinear Schrödinger equation is given.