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

### Stefan Le Coz

Université Paul Sabatier, Toulouse, France### Yvan Martel

École Polytechnique, Palaiseau, France### Pierre Raphaël

Université de Nice Sophia Antipolis, France

## Abstract

We consider a nonlinear Schrödinger equation with double power nonlinearity

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.

## 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. 795–833

DOI 10.4171/RMI/899