A Fourier approach for nonlinear equations with singular data
type: Item
instance of: scholarly article
title: A Fourier approach for nonlinear equations with singular data
zbMATH Open document ID: 1281.35037
author: Lucas C. F. Ferreira
author: Marcelo Montenegro
published in: Israel Journal of Mathematics
publication date: 2013-06-19T00:00:00Z
review text: For \(0<m<n\), \(p\) a positive integer and \(p>n/(n-m)\), the authors study the inhomogeneous equation \(Lu+u^p+V(x)u+f(x)=0\) in \({\mathbb R}^n\) with singular data \(f\) and \(V\). The symbol \(\sigma\) of the operator \(L\) is bounded from below by \(|\xi|^m\). Examples of \(L\) are Laplacian, biharmonic and fractional order operators. Here, \(f\) and \(V\) can have infinite singular points, change sign, oscillate at infinity, and be measures. Also, \(f\) and \(V\) can blow up on an unbounded \((n-1)\)-manifold. The solution \(u\) can change sign, be nonradial and singular. If \(\sigma\), \(f\) and \(V\) are radial, then \(u\) is radial. The assumptions of \(f\) and \(V\) are in terms of their Fourier transform, and some examples are provided. Usually, the Fourier transform is a powerful tool to solve linear partial differential equations, because the resulting algebraic equation can be solved for \(\hat{u}\). And by inversion, \({\hat{u}}\) turns out to be a solution of the original equation. In this paper, the authors successfully carry out a resembling strategy for a nonlinear equation and the corresponding nonalgebraic equation in Fourier variables.
reviewed by: Ravshan Ashurov
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35J75
Mathematics Subject Classification ID: 35A22
zbMATH DE Number: 6176903
zbMATH Keywords: differential operators
zbMATH Keywords: Laplacian
zbMATH Keywords: singular data
zbMATH Keywords: nonlinear equations
zbMATH Keywords: existence and uniqueness
zbMATH Keywords: decay and regularity
zbMATH Keywords: Fourier transforms
MaRDI profile type: Publication
full work available at URL: s11856-012-0032-1
OpenAlex ID: W2089322962
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DOI: 10.1007/S11856-012-0032-1
description: scientific article; zbMATH DE number 6176903