Existence and instability of standing waves for the biharmonic nonlinear Schroedinger equation with combined nonlinearities

Abstract

We prove the existence of normalized ground state solutions for the biharmonic Schr\"odinger equation with combined nonlinearities and show that all ground states correspond to the local minima of the associated energy functional restricted to the appropriate set. Moreover, we prove that the standing waves are strongly unstable by blowup. In particular, our results cover the critical case. Our method is novel and innovative as previous ideas cannot be used in many cases under this study.