Construction of pseudo-gradient vector field and sign-changing multiple solutions involving \(p\)-Laplacian
type: Item
instance of: scholarly article
title: Construction of pseudo-gradient vector field and sign-changing multiple solutions involving \(p\)-Laplacian
zbMATH Open document ID: 1079.35035
author: J. Chen
author: Zhitao Zhang
author: Shujie Li
published in: Journal of Differential Equations
publication date: 2004-08-10T00:00:00Z
review text: In the present study the authors consider the existence of multiple and sign-changing solutions of the problem \[ -\Delta_p u=f(u)\quad \text{in }\Omega, \qquad u(x)=0,\quad \text{on }\partial\Omega, \tag{1} \] where \(-\Delta_p u:= \text{div}(|\nabla u|^{p-2}\nabla u)\) is the \(p\)-Laplacian, \(1< p<+\infty\), \(\Omega\) is a smooth bounded domain in \(\mathbb R^N\) \((N\geq 1)\), \(f(u)\) is the local Lipschitz continuous having ``jumping'' nonlinearities at zero or infinity. To this end, the authors carefully construct a pseudo-gradient vector field in \(C_0^1 (\overline{\Omega})\), by which they obtain the positive and negative cones in \(C_0^1 (\overline{\Omega})\), which are all the invariant sets of the descent flow of the corresponding functional and use (PS) condition.
Mathematics Subject Classification ID: 35J20
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 47J30
zbMATH DE Number: 2085854
zbMATH Keywords: critical points
zbMATH Keywords: multiple solutions
zbMATH Keywords: sign-changing solution
zbMATH Keywords: \(p\)-Laplacian
reviewed by: Messoud Efendiev
MaRDI profile type: Publication
full work available at URL: j.jde.2004.03.019
OpenAlex ID: W2018723842
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DOI: 10.1016/J.JDE.2004.03.019
description: scientific article; zbMATH DE number 2085854