Multi-bump solutions for logarithmic Schrödinger equations
type: Item
instance of: scholarly article
title: Multi-bump solutions for logarithmic Schrödinger equations
zbMATH Open document ID: 1383.35217
author: Kazunaga Tanaka
author: C. X. Zhang
published in: Calculus of Variations and Partial Differential Equations
publication date: 2017-05-15T00:00:00Z
full work available at URL: https://arxiv.org/abs/1608.01742
review text: The paper addresses the stationary version of the nonlinear Schrödinger equation for real function \(u(x)\) with the log nonlinearity in the \(N\)-dimensional space, with 1-periodic coefficient functions, \(V(x)\) and \(Q(x)\): \[ -\Delta u +V(x)u=Q(x)u\ln(u^2). \] This equation with constant coefficients \(V\) and \(Q\) gives rise to the well-known exact solutions in the form of \textit{gaussons}. This paper produces a proof of the existence of infinity many multi-peak solutions of the equation with the \(x\)-periodic coefficients. The solutions are looked for as stationary points of the respective Hamiltonian. The logarithmic singularity in the Hamiltonian, corresponding to the log nonlinearity of the equation, makes it necessary to address the fixed-point problem admitting its lack of smoothness.
Mathematics Subject Classification ID: 35Q55
Mathematics Subject Classification ID: 81Q05
zbMATH DE Number: 6715632
zbMATH Keywords: fixed point
zbMATH Keywords: nonlinear Schrödinger equation
zbMATH Keywords: Hamiltonian
reviewed by: B. A. Malomed
MaRDI profile type: Publication
OpenAlex ID: W2963781133
arXiv ID: 1608.01742
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DOI: 10.1007/S00526-017-1122-Z
description: scientific article