Arbitrarily many solutions for a perturbed \(p(x)\)-Laplacian equation involving oscillatory terms
type: Item
instance of: scholarly article
title: Arbitrarily many solutions for a perturbed \(p(x)\)-Laplacian equation involving oscillatory terms
zbMATH Open document ID: 1178.35172
author: Guowei Dai
published in: Applied Mathematics and Computation
publication date: 2009-12-18T00:00:00Z
review text: This paper deals with a perturbed \(p(x)\)-Laplacian equation involving oscillatory terms: \[ \begin{cases} -\text{div}\left(|\nabla u|^{p(x)-2}\nabla u\right)+|u|^{p(x)-2}u=Q(x)[f(u)+\varepsilon g(u)]x\in\mathbb R^N,\\u\geq0,\\u(x)\to 0\text{ as }|x|\to\infty. \end{cases} \] By using variational methods and the non-smooth version of the symmetric criticality principle, we establish (a) the unperturbed problem \(P_0\) has infinitely many distinct solutions; (b) the number of distinct solutions for \(P_\varepsilon\) becomes greater whenever \(\varepsilon\) becomes smaller. Our results are a generalization of the case of the Laplacian from \textit{A. Kristály} [J. Differ. Equations 245, No. 12, 3849--3868 (2008; Zbl 1169.35025)] to the case of the \(p(x)\)-Laplacian.
Mathematics Subject Classification ID: 35J62
Mathematics Subject Classification ID: 35J92
Mathematics Subject Classification ID: 35J20
Mathematics Subject Classification ID: 35B20
zbMATH DE Number: 5649904
zbMATH Keywords: \(p(x)\)-Laplacian
zbMATH Keywords: oscillatory terms
zbMATH Keywords: perturbed elliptic problem
MaRDI profile type: Publication
full work available at URL: j.amc.2009.08.049
OpenAlex ID: W1983222109
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DOI: 10.1016/J.AMC.2009.08.049
description: scientific article; zbMATH DE number 5649904