On stability and instability of standing waves for the inhomogeneous fractional Schrodinger equation
In this paper, we consider the stability and instability of standing waves for the inhomogeneous fractional Schrodinger equation i\partial_t\psi= \Delta)^s\psi |x| b}\psi|{2p}\psi. By applying the profile decomposition of bounded sequences in $H^s$ and variational methods, in the $L^2$ subcritical case, i.e. $0 p \frac{4s 2b}{N} we prove that the standing waves are orbitally stable. In the $L^2$ critical case, i.e. $p=\frac{4s 2b}{N} we show that the standing waves are strongly unstable by blow up.