Construction of solutions for some localized nonlinear Schrödinger equations

Olivier Bourget; Matias Courdurier; Claudio Fernández

doi:10.3934/dcds.2019035

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2019, Volume 39, Issue 2: 841-862. Doi: 10.3934/dcds.2019035

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Construction of solutions for some localized nonlinear Schrödinger equations

Departamento de Matemática, Pontificia Universidad Católica de Chile, Av. Vicuña Mackenna 4860, Santiago, Chile

* Corresponding author: mcourdurier@mat.uc.cl
* Corresponding author: mcourdurier@mat.uc.cl

Received: December 31, 2017

Revised: June 30, 2018

Published: January 2019

O. B. was partially supported by FONDECYT grant number 1161732. M. C. was partially supported by FONDECYT grant number 1141189. C. F. was partially supported by FONDECYT grant number 1141120

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Abstract

For an $N$-body system of linear Schrödinger equation with space dependent interaction between particles, one would expect that the corresponding one body equation, arising as a mean field approximation, would have a space dependent nonlinearity. With such motivation we consider the following model of a nonlinear reduced Schrödinger equation with space dependent nonlinearity

$\begin{align*}-\varphi''+V(x)h'(|\varphi|^2)\varphi = λ \varphi,\end{align*}$

where $V(x) = -χ_{[-1,1]} (x)$ is minus the characteristic function of the interval $[-1,1]$ and where $h'$ is any continuous strictly increasing function. In this article, for any negative value of $λ$ we present the construction and analysis of the infinitely many solutions of this equation, which are localized in space and hence correspond to bound-states of the associated time-dependent version of the equation.

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Mathematics Subject Classification: Primary: 35J60, 35P30; Secondary: 35C08.

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