Periodic solutions of a wave equation with concave and convex nonlinearities
type: Item
instance of: scholarly article
title: Periodic solutions of a wave equation with concave and convex nonlinearities
zbMATH Open document ID: 0926.35013
author: Yanheng Ding
author: Thomas J. Bartsch
author: C. Lee
published in: Journal of Differential Equations
publication date: 1999-05-31T00:00:00Z
review text: The paper deals with the following problem describing the periodic solutions of a nonautonomous wave equation \[ u_{tt}-u_{xx}=f_{\lambda\mu}(x,t,u), \quad\text{for }0<x<\pi,\;t\in \mathbb{R}, \] \[ u(0,t)=u(\pi,t)=0,\quad \text{for }t\in \mathbb{R},\qquad u(x,t+2\pi)=u(x,t),\quad \text{for }t\in \mathbb{R}. \] The nonlinearity \(f_{\lambda\mu}\in C([0,\pi]\times \mathbb{R}^2,\mathbb{R})\) has the form \(f_{\lambda\mu}=\lambda g(x,t,\xi)+\mu h(x,t,\xi)\) with \(\lambda,\mu\in \mathbb{R}\), \(g\) superlinear in \(\xi\), \(h\) sublinear in \(\xi\) and both \(g\), \(h\) \(2\pi\)-periodic in \(t\) and nondecreasing in \(\xi\). If \(\lambda\) and \(\mu\) have the same sign then \(f_{\lambda\mu}\) is monotone. The paper treats also the case \(\lambda\mu<0\) where this monotonicity is lost. Using variational methods the existence and multiplicity of periodic solutions of this problem are studied for the model nonlinearity \(f_{\lambda\mu}=a(x,t) (\lambda | \xi| ^{p-2}\xi+\mu| \xi| ^{q-2}\xi) \) with \(p>2>q>1\) and \(a\in C([0,\pi]\times \mathbb{R},\mathbb{R})\) positive and \(2\pi\)-periodic in \(t\) and in the general case.
Mathematics Subject Classification ID: 35B10
Mathematics Subject Classification ID: 35L70
Mathematics Subject Classification ID: 35L20
zbMATH DE Number: 1278769
zbMATH Keywords: existence and multiplicity of periodic solutions
zbMATH Keywords: variational methods
reviewed by: Ivan Ginchev
MaRDI profile type: Publication
full work available at URL: jdeq.1998.3551
OpenAlex ID: W2912620585
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DOI: 10.1006/JDEQ.1998.3551
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description: scientific article; zbMATH DE number 1278769