Subharmonic solutions for nonautonomous sublinear second-order differential inclusions systems with \(p\)-Laplacian
type: Item
instance of: scholarly article
title: Subharmonic solutions for nonautonomous sublinear second-order differential inclusions systems with \(p\)-Laplacian
zbMATH Open document ID: 1160.34008
author: Daniel Paşca
author: Chun-Lei Tang
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2008-09-10T00:00:00Z
review text: Consider the differential inclusion \[ -\frac{d}{dt}(\| \dot{u}(t)\| ^{p-2}\dot{u}(t))\in\partial F(t,u(t))\quad \text{ a.e. }\, t\in \mathbb{R}, \tag{1} \] where \(p>1, F:\mathbb{R}\times\mathbb{R}^{n}\to \mathbb{R}\) is \(T\)-periodic (\(T>0\)) in \(t\) for all \(x\in\mathbb{R}^{n}\), and \(\partial\) denotes the Clarke subdifferential. \newline Theorem. Let \(F(t,x)\) be integrable in \(t\) over \([0,T]\) for each \(x\in \mathbb{R}^{n}\) and is locally Lipschitz in \(x\) for each \(t\in [0,T].\) It is supposed that the following conditions hold: \newline (i)\, there exist \(c_{1},c_{2} >0\) and \(\alpha\in[0,p-1)\) such that \(\zeta\in\partial F(t,x)\Rightarrow \| \zeta\| \leq c_{1}\| x\| ^{\alpha} +c_{2} \) for all \(x\in\mathbb{R}^{n}\) and a.e. \(t\in[0,T];\) \newline (ii)\, there exists \(\gamma\in L^{1}(0,T)\) such that \(F(t,x)\geq\gamma(t)\) for all \(x\in\mathbb{R}^{n}\) and a.e. \(t\in[0,T];\) \newline (iii)\, there exists a subset \(E\) of \([0,T]\) with meas \((E)>0\) such that \(\| x\| ^{-q\alpha}F(t,x)\to +\infty\) as \(\| x\| \to \infty\) for a.e. \(t\in E\), where \(q= p/(p-1).\) \newline Then the differential inclusion (1) has a \( kT\)-periodic solution \(u_{k}\in W^{1,p}_{kT}\) for every positive integer \(k\) such that \(\| u_{k}\| _{\infty}\to\infty\) as \(k\to\infty\).
Mathematics Subject Classification ID: 34A60
Mathematics Subject Classification ID: 34C25
zbMATH DE Number: 5323975
zbMATH Keywords: \(p\)-Laplacian, differential inclusions systems, subharmonic solutions
zbMATH Keywords: critical point, sublinear
MaRDI profile type: Publication
full work available at URL: j.na.2007.06.019
OpenAlex ID: W2039523522
cites work: Variational methods for non-differentiable functionals and their applications to partial differential equations
cites work: Q3996569
cites work: Critical point theory and Hamiltonian systems
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DOI: 10.1016/J.NA.2007.06.019
description: scientific article; zbMATH DE number 5323975