We consider a nonlinear Schrödinger equation with double power nonlinearity
in (). Classical variational arguments ensure that data with lead to global in time solutions, where is the ground state of the mass critical problem (). We are interested by the threshold dynamic and in particular by the existence of finite time blow up minimal solutions. For , such an object exists thanks to the explicit conformal symmetry, and is in fact unique from the seminal work . For , simple variational arguments ensure that minimal mass data lead to global in time solutions. We investigate in this paper the case , exhibiting a new class of minimal blow up solutions with blow up rates deeply affected by the double power nonlinearity. The analysis adapts the recent approach  for the construction of minimal blow up elements.
Cite this article
Stefan Le Coz, Yvan Martel, Pierre Raphaël, Minimal mass blow up solutions for a double power nonlinear Schrödinger equation. Rev. Mat. Iberoam. 32 (2016), no. 3, pp. 795–833DOI 10.4171/RMI/899