Existence results for a perturbed problem involving fractional Laplacians
type: Item
instance of: scholarly article
title: Existence results for a perturbed problem involving fractional Laplacians
zbMATH Open document ID: 1475.35387
DOI: 10.1155/2014/548301
author: Yan Hu
published in: Abstract and Applied Analysis
publication date: 2019-02-14T00:00:00Z
review text: Summary: We extend the results of \textit{X. Cabré} and \textit{Y. Sire} [Trans. Am. Math. Soc. 367, No. 2, 911--941 (2015; Zbl 1317.35280), \url{arXiv:1111.0796}] to show the existence of layer solutions of fractional Laplacians with perturbed nonlinearity \((- \Delta)^s u = b(x) f(u)\) in \(\mathbb{R}\) with \(s \in(0,1)\). Here \(b\) is a positive periodic perturbation for \(f\), and \(- f\) is the derivative of a balanced well potential \(G\). That is, \(G \in C^{2, \gamma}\) satisfies \(G(1) = G(- 1) < G(\tau) \forall \tau \in(- 1,1), G'(1) = G'(- 1) = 0 \). First, for odd nonlinearity \(f\) and for every \(s \in(0,1)\), we prove that there exists a layer solution via the monotone iteration method. Besides, existence results are obtained by variational methods for \(s \in(1 / 2, 1)\) and for more general nonlinearities. While the case \(s \leq 1 / 2\) remains open.
Mathematics Subject Classification ID: 35R11
Mathematics Subject Classification ID: 35A01
Mathematics Subject Classification ID: 35J91
zbMATH DE Number: 7022604
Wikidata QID: Q59039396
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2014/548301
OpenAlex ID: W2056293810
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description: scientific article; zbMATH DE number 7022604