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

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 $\Vert u_0{\Vert }_2<\Vert Q{\Vert }_2$ lead to global in time solutions, where $Q$ is the ground state of the mass critical problem ($ϵ=0$). We are interested by the threshold dynamic $\Vert u_0{\Vert }_2=\Vert Q{\Vert }_2$ and in particular by the existence of finite time blow up minimal solutions. For $ϵ=0$, such an object exists thanks to the explicit conformal symmetry, and is in fact unique from the seminal work [22]. For $ϵ=-1$, simple variational arguments ensure that minimal mass data lead to global in time solutions. We investigate in this paper the case $ϵ=1$, 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.