Problems at resonance for equations with periodic nonlinearities.
type: Item
instance of: scholarly article
title: Problems at resonance for equations with periodic nonlinearities.
zbMATH Open document ID: 1046.34034
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2003-11-10T00:00:00Z
review text: The results in this article concern differential equations with periodic nonlinearities. In the first part, the authors prove an abstract result concerning some equations involving selfadjoint elliptic operators. The method is variational, using a saddle point theorem and properties of perturbations of functionals having saddle points. This result is then applied to the equation \[ -u''-u+g(u)=h(t)+a\sin(t-\phi), \tag{*} \] where \(g\) is a \(C^1\) periodic function with zero mean value and \(h\) is orthogonal to \(\cos t\) and to \(\sin t\). For periodic boundary conditions (for \(u\) and \(u'\)) and \(a\) small enough, the existence of \(2\pi\)-periodic solutions is proved. Also, an application of the abstract result to a Dirichlet boundary value problem on a square is presented. Using the same strategy, the authors obtain multiple solutions for equation \((*)\) in the case \(g(u)=-\sin(u)\).
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 35B34
Mathematics Subject Classification ID: 35J20
zbMATH DE Number: 2001960
zbMATH Keywords: periodic solution
zbMATH Keywords: variational method
zbMATH Keywords: elliptic operator
author: Denis Bonheure
author: Christian Fabry
author: David Ruiz
reviewed by: Gabriela Liţcanu
MaRDI profile type: Publication
full work available at URL: j.na.2003.07.005
OpenAlex ID: W2024877561
cites work: Dual variational methods in critical point theory and applications
cites work: Periodic solutions of Hamiltonian systems with strong resonance at infinity
cites work: Critical point theorems and applications to nonlinear boundary value problems
cites work: Abstract critical point theorems and applications to some nonlinear problems with “strong” resonance at infinity
cites work: Existence and multiplicity of solutions of some conservative pendulum-type equations with homogeneous Dirichlet conditions
cites work: Resonant problems with multidimensional kernel and periodic nonlinearities.
cites work: Nonlinear problems with strong resonance at infinity: an abstract theorem and applications
cites work: A strong resonance problem
cites work: Stationary Phase in Two Dimensions
cites work: On the use of asymptotics in nonlinear boundary value problems
cites work: Q5185568
cites work: On the variational principle
cites work: Q3232397
cites work: Q4272029
cites work: Semilinear Sturm-Liouville problem with periodic nonlinearity
cites work: A note on a resonance problem
cites work: Critical point theory and Hamiltonian systems
cites work: Q3215858
cites work: On the solvability of some elliptic partial differential equations with the linear part at resonance
cites work: T-periodic solutions of time dependent Hamiltonian systems with a potential vanishing at infinity
cites work: A boundary value problem with a periodic nonlinearity
DOI: 10.1016/J.NA.2003.07.005
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description: scientific article; zbMATH DE number 2001960