Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation

Alex H. Ardila; Mykael Cardoso

doi:10.3934/cpaa.2020259

Article Contents

2021, Volume 20, Issue 1: 101-119. Doi: 10.3934/cpaa.2020259

This issue Previous Article Scattering of the focusing energy-critical NLS with inverse square potential in the radial case Next Article Uniqueness and nondegeneracy of ground states for nonlinear Schrödinger equations with attractive inverse-power potential

Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation

Alex H. Ardila^1, , and
Mykael Cardoso^2,

1.
Department of Mathematics, ICEx-UFMG, Universidade Federal de Minas Gerais-ICEx, Caixa Postal 702, CEP 30123-970, Belo Horizonte-MG, Brazil
2.
Department of Mathematics, CCN, Universidade Federal do Piauí, Ininga - CEP: 64049-550, Teresina - PI, Brasil

^* Corresponding author
^* Corresponding author

Received: April 2020

Revised: August 2020

Early access: October 2020

Published: January 2021

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Abstract

Using variational methods we study the stability and strong instability of ground states for the focusing inhomogeneous nonlinear Schrödinger equation (INLS)

$ \begin{equation*} i\partial_{t}u+\Delta u+|x|^{-b}|u|^{p-1}u = 0. \end{equation*} $

We construct two kinds of invariant sets under the evolution flow of (INLS). Then we show that the solution of (INLS) is global and bounded in $ H^{1}(\mathbb{R^{N}}) $ in the first kind of the invariant sets, while the solution blow-up in finite time in the other invariant set. Consequently, we prove that if the nonlinearity is $ L^{2} $-supercritical, then the ground states are strongly unstable by blow-up.

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Mathematics Subject Classification: Primary: 35Q55; Secondary: 35B44.

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