Minimal mass blow up solutions for a double power nonlinear Schr\"odinger equation

TL;DR

This paper constructs a new class of minimal mass blow-up solutions for a double power nonlinear Schrödinger equation, revealing how the interplay of focusing and sub-critical nonlinearities affects blow-up dynamics.

Contribution

It introduces minimal blow-up solutions for a double power NLS with novel blow-up rates influenced by both nonlinearities, extending recent methods to this complex setting.

Findings

Existence of minimal blow-up solutions with new blow-up rates

Blow-up behavior deeply affected by double power nonlinearity

Extension of recent construction methods to complex nonlinearities

Abstract

We consider a nonlinear Schr\"odinger 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 Rapha\"el and Szeftel for the construction of minimal blow up elements.