Existence of a positive solution to Kirchhoff problems involving the fractional Laplacian
type: Item
instance of: scholarly article
title: Existence of a positive solution to Kirchhoff problems involving the fractional Laplacian
zbMATH Open document ID: 1327.35401
DOI: 10.4171/ZAA/1547
author: Bin Ge
author: Chao Zhang
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 2015-11-17T00:00:00Z
review text: Summary: The goal of this paper is to establish the existence of a positive solution to the following fractional Kirchhoff-type problem \[ \left(1+\lambda\int_{\mathbb{R}^N} \left( \big|(-\Delta)^{\frac{\alpha}{2}}u(x) \big|^{2}+V(x)u^2 \right)dx \right)\big[(-\Delta)^\alpha u+V(x)u\big]=f(u) \quad \text{in } \mathbb{R}^N, \] where \(N\geq 2\), \(\lambda\geq 0\) is a parameter, \(\alpha\in(0,1)\), \((-\Delta)^\alpha\) stands for the fractional Laplacian, \(f\in C(\mathbb{R}_+,\mathbb{R}_+)\). Using a variational method combined with suitable truncation techniques, we obtain the existence of at least one positive solution without compactness conditions.
Mathematics Subject Classification ID: 35R11
Mathematics Subject Classification ID: 35A15
Mathematics Subject Classification ID: 35A01
Mathematics Subject Classification ID: 35B09
Mathematics Subject Classification ID: 35R09
zbMATH DE Number: 6510072
zbMATH Keywords: fractional-Laplacian
zbMATH Keywords: variational method
zbMATH Keywords: cut-off function
zbMATH Keywords: Pohozaev type identity
MaRDI profile type: Publication
cites work: Elliptic problems involving the fractional Laplacian in \(\mathbb R^N\)
cites work: Ground state of scalar field equations involving a fractional Laplacian with general nonlinearity
cites work: Classification of solutions for an integral equation
cites work: Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian
full work available at URL: https://doi.org/10.4171/zaa/1547
OpenAlex ID: W1845060646
description: scientific article; zbMATH DE number 6510072