(Super)critical nonlocal equations with periodic boundary conditions
type: Item
instance of: scholarly article
title: (Super)critical nonlocal equations with periodic boundary conditions
zbMATH Open document ID: 1404.35194
author: Vincenzo Ambrosio
author: Jean Mawhin
author: Giovanni Molica Bisci
published in: Selecta Mathematica. New Series
publication date: 2018-09-21T00:00:00Z
review text: In this paper the authors discuss the existence and multiplicity of periodic solutions for a class of parametric nonlocal equations with critical and supercritical growth. More specifically, given \(s\in (0,1)\), \(N>2s\), \(m>0\), the authors investigate the existence and multiplicity of periodic solutions for the class of problems \[ ((-\Delta+m^2)^s)u=\lambda f(x,u)+\mu g(x,u) \text{ in } (0,T)^N, \] \[ u(x+Te_i)=u(x) \text{ for all } x\in \mathbb{R}^N, \;i=1,\ldots, N. \] Here \(f\) has subcritical growth and \(g\) behaves as \(|u|^{p-2}u\), with \(p\geq 2N/(N-2s)\), besides satisfying additional conditions. The first key ingredient in the proof is an extension technique in the periodic setting, inspired by the one of Caffarelli-Silvestre, which realizes the equations as local degenerate elliptic problems in a half-cylinder of \(\mathbb{R}^{N+1}_+\) with nonlinear Neumann boundary conditions. The authors then truncate the critical term \(g\) and prove that the critical points of the truncated functional are bounded. The Moser iteration scheme adapted to the fractional setting is also instrumental. Finally, a multiplicity result valid for differentiable functionals due to Ricceri is used, and the authors show that the problem under consideration admits at least three period solutions with the property that their Sobolev norms are bounded by a suitable constant. Additionally, the authors also provide a concrete estimate on the range of these parameters by using properties of fractional calculus.
Mathematics Subject Classification ID: 35J70
Mathematics Subject Classification ID: 35R11
Mathematics Subject Classification ID: 35B10
Mathematics Subject Classification ID: 35A15
zbMATH DE Number: 6941791
zbMATH Keywords: fractional operators
zbMATH Keywords: multiple periodic solutions
reviewed by: Mariana Vega Smit
MaRDI profile type: Publication
full work available at URL: s00029-018-0398-y
OpenAlex ID: W2789776284
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(Super)critical nonlocal equations with periodic boundary conditions (DOI)
DOI: 10.1007/S00029-018-0398-Y