Revista Matemática Iberoamericana

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Volume 32, Issue 3, 2016, pp. 795–833
DOI: 10.4171/RMI/899

Published online: 2016-10-03

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

Stefan Le Coz[1], Yvan Martel[2] and Pierre Raphaël[3]

(1) Université Paul Sabatier, Toulouse, France
(2) École Polytechnique, Palaiseau, France
(3) Université de Nice Sophia Antipolis, France

We consider a nonlinear Schrödinger equation with double power nonlinearity \[ i\partial_tu +\Delta u+|u|^{4/d}u+\epsilon |u|^{p-1}u=0, \quad \epsilon\in\{-1,0,1\}, \quad 1 < p <1 + \frac 4d \] in $\mathbb R^d$ ($d=1,2,3$). Classical variational arguments ensure that $H^1(\mathbb R^d)$ data with $\|{u_0}\|_{2}<\|{Q}\|_{2}$ lead to global in time solutions, where $Q$ is the ground state of the mass critical problem ($\epsilon=0$). We are interested by the threshold dynamic $\|{u_0}\|_{2}=\|{Q}\|_{2}$ and in particular by the existence of finite time blow up minimal solutions. For $\epsilon=0$, such an object exists thanks to the explicit conformal symmetry, and is in fact unique from the seminal work [22]. For $\epsilon=-1$, simple variational arguments ensure that minimal mass data lead to global in time solutions. We investigate in this paper the case $\epsilon=1$, 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 [31] for the construction of minimal blow up elements.

Keywords: Blow-up, nonlinear Schrödinger equation, double power nonlinearity, minimal mass, critical exponent

Le Coz Stefan, Martel Yvan, Raphaël Pierre: Minimal mass blow up solutions for a double power nonlinear Schrödinger equation. Rev. Mat. Iberoam. 32 (2016), 795-833. doi: 10.4171/RMI/899