Some qualitative studies of the focusing inhomogeneous Gross-Pitaevskii equation

We study the Cauchy problem for an inhomogeneous Gross-Pitaevskii equation. We first derive a sharp threshold for global existence and blow up of the solution. Then we construct and classify finite time blow up solutions at the minimal mass threshold. Additionally, using variational techniques, we study the existence, the orbital stability and instability of standing waves.