Minimal mass blow up solutions for a double power nonlinear Schrödinger equation

Stefan Le Coz (1) , Yvan Martel (2) , Pierre Raphaël (3)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 CMLS - Centre de Mathématiques Laurent Schwartz
3 LJAD - Laboratoire Jean Alexandre Dieudonné

We consider a nonlinear Schrödinger equation with double power nonlinearity, where one power is focusing and mass critical and the other mass sub-critical. Classical variational arguments ensure that initial data with mass less than the mass of the ground state of the mass critical problem lead to global in time solutions. We are interested by the threshold dynamic and in particular by the existence of finite time blow up minimal solutions. For the mass critical problem, such an object exists thanks to the explicit conformal symmetry, and is in fact unique. For the focusing double power nonlinearity, we exhibit 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 developed by Raphaël and Szeftel for the construction of minimal blow up elements.

Mots clés: Nonlinear Schrödinger equations; blow-up; double power nonlinearity; minimal mass; critical exponent

Domaines: Equations aux dérivées partielles [math.AP]

Stefan Le Coz, Yvan Martel, Pierre Raphaël. Minimal mass blow up solutions for a double power nonlinear Schrödinger equation. Revista Matemática Iberoamericana, 2016, 32 (3), pp.795-833. ⟨10.4171/rmi/899⟩. ⟨hal-01011271⟩