Boundedness of solutions for equations with \(p\)-Laplacian and an asymmetric nonlinear term
type: Item
instance of: scholarly article
title: Boundedness of solutions for equations with \(p\)-Laplacian and an asymmetric nonlinear term
zbMATH Open document ID: 1069.34052
author: Bin Liu
published in: Journal of Differential Equations
publication date: 2005-01-11T00:00:00Z
review text: Sufficient conditions are obtained for the boundedness of solutions and existence of quasi-periodic solutions of differential equations of the form \[ (\phi_p(x'))'+\alpha\phi_p(x^+)-\beta\phi_p(x^-)=f(t,x),\tag{\(*\)} \] where \(x^+=\max\{x,0\}\), \(x^-=\max\{-x,0\}\), \(\phi_p(s)=|s|^{p-2}s,p>1\) and \(\alpha,\beta\) are strictly positive constants, \(f\) is smooth and \(2\pi\) periodic in \(t\). The author has reviewed the development of the work in this direction leading to the present study. The main result of the paper runs as follows: Suppose that \(f\in C^{7,6}(S^1\times \mathbb{R})\), \(\lim_{x\to\pm\infty}f(t,x)=f_\pm(t)\), uniformly in \(t\) and \(\lim_{x\to\pm\infty}x^m\frac{\partial^{n+m}}{\partial t^n\partial x^m}f(t,x)=f_{\pm,m,n}(t)\), for \((n,m)=(0,6)\), (7,0) and (7,6). Let \(f_{\pm,m,n}(t)\equiv 0\) for \(m=6, n=0,7\). If \[ Z(\theta)=\int_{\{t\in [0,\frac{2\pi}{n}]: \nu(t)>0\}} f_+(t+\theta)\nu(t)dt +\int_{\{t\in [0,\frac{2\pi}{n}]: \nu(t)<0\}} f_-(t+\theta)\nu(t)dt \] is of constant sign, then all solutions of (\(*\)) are defined in \(\mathbb{R}\) and for each \(x(t)\), we have \(\sup_{t\in \mathbb{R}}\{|x(t)|+|x'(t)|\}<+\infty\). Moreover, (\(*\)) admits infinitely many subharmonic solutions and quasi-periodic solutions. It is a good paper.
Mathematics Subject Classification ID: 34C11
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 34C27
zbMATH DE Number: 2126469
zbMATH Keywords: boundedness of solutions
zbMATH Keywords: \(p\)-Laplacian
zbMATH Keywords: small twist theorem
MaRDI profile type: Publication
full work available at URL: j.jde.2004.06.023
OpenAlex ID: W2090334779
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DOI: 10.1016/J.JDE.2004.06.023
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description: scientific article; zbMATH DE number 2126469