Stationary sign changing solutions for an inhomogeneous nonlocal problem
type: Item
instance of: scholarly article
title: Stationary sign changing solutions for an inhomogeneous nonlocal problem
zbMATH Open document ID: 1245.45001
author: Carmen Cortázar
author: Manuel Elgueta
author: Salomé Martínez
author: Jorge García-Melian
published in: Indiana University Mathematics Journal
publication date: 2012-05-30T00:00:00Z
full work available at URL: http://hdl.handle.net/10533/138599
full work available at URL: http://www.iumj.indiana.edu/IUMJ/ABS/2011/4385
Mathematics Subject Classification ID: 45E10
Mathematics Subject Classification ID: 45A05
zbMATH DE Number: 6039333
zbMATH Keywords: nonlocal diffusion
zbMATH Keywords: sign changing solution
zbMATH Keywords: uniqueness
zbMATH Keywords: singular integral equation
MaRDI profile type: Publication
OpenAlex ID: W44930369
DOI: 10.1512/IUMJ.2011.60.4385
review text: The authors prove the existence of a sign changing solution \(q(x)\) of the nonlocal equation NEWLINE\[NEWLINE \int\limits_R J\left(\frac{x-y}{g(y)}\right)\frac{u(y)}{g(y)}dy - u(x) = 0, \, x \in R, \tag{1} NEWLINE\]NEWLINE where \(J\) is an even, compactly supported, Holder continuous probability kernel, \(g\) is a continuous function, bounded and bounded away from zero in \(R.\) The solution \(q(x)\) has the following properties: 1) it is strictly positive when \(x > k\) and strictly negative for \(x < -k,\) provided that \(k\) is chosen large enough, 2) the solution \(q(x)\) verifies the inequalities \(a_1 \leq q(x)/x \leq a_2\) for positive constants \(a_1,\) \(a_2\) and large \(|x|.\) The authors investigate the properties of solutions with exponensial and potential growth.
reviewed by: Ilia V. Boikov
description: scientific article; zbMATH DE number 6039333