Strong instability of standing waves for nonlinear Schr\"odinger   equations with a delta potential
  We study strong instability (instability by blowup) of standing wave solutions for a nonlinear Schr\"odinger equation with an attractive delta potential and $L^2$-supercritical power nonlinearity in one space dimension. We also compare our sufficient condition on strong instability with some known results on orbital instability. 
Masahito Ohta and Takahiro Yamaguchi
math.AP