The authors announce the following result on nonlinear Schrödinger equations of the form \[ i\phi_ t-\Delta \phi =| \phi |^{p- 1}\phi \quad with\quad 1+(4/N)\leq p<(N+2)/(N-2) \] in \({\mathbb{R}}^ N:\) The standing wave solutions \(e^{i\omega t} u(x)\) are unstable solutions of the equations, i.e. there exist initial data arbitrarily close to the standing waves for which the solution of the corresponding Cauchy problem blows up in finite time. A sketch is given of the proof of the announced result which uses a constructive approach to critical point theory and deformation arguments.