A remark on the existence of multiple solutions to a multiparameter nonlinear elliptic system
type: Item
instance of: scholarly article
title: A remark on the existence of multiple solutions to a multiparameter nonlinear elliptic system
zbMATH Open document ID: 1173.35444
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2009-06-09T00:00:00Z
review text: In this work, motivated by [\textit{T. F. Wu}, Nonlinear Anal., Theory Methods Appl. 68, No.~6 (A), 1733--1745 (2008; Zbl 1151.35342)], and using recent ideas from \textit{K. J. Brown} and \textit{T. F. Wu} [J. Math. Anal. Appl. 337, No.~2, 1326--1336 (2008; Zbl 1132.35361)], we prove the existence of nontrivial nonnegative solutions to the nonlinear elliptic system \[ \begin{cases} -\Delta_pu+m(x)|u|^{p-2}u=\lambda|u|^{\gamma-2}u+\frac{\alpha}{\alpha+\beta}\;c(x)|u|^{\alpha-2}u|v|^\beta,\quad & x\in\Omega,\\ -\Delta_pv+m(x)|v|^{p-2}v=\lambda|v|^{\gamma-2}v+\frac{\alpha}{\alpha+\beta}\;c(x)|u|^{\alpha-2}u|v|^\beta,& x\in\Omega,\\ u=0=v, & x\in\partial\Omega.\end{cases} \] Here \(\Delta_p\) denotes the \(p\)-Laplacian operator defined by \(\Delta_pz=\text{div}(|\nabla z|^{p-2}\nabla z)\), \(p>2\), \(\Omega\subset \mathbb R^N\) is a bounded domain with smooth boundary, \(\alpha>1\), \(\beta>1\), \(2<\alpha+\beta<p<\gamma<p*\) \((p^*=\frac{pN}{N-p}\) if \(N>p\), \(p^*=\infty\) if \(N\leq p\)), \(\frac{\partial}{\partial n}\) is the outer normal derivative, \((\lambda,\mu)\in\mathbb R^2\setminus \{(0,0)\}\), the weight \(m(x)\) is a bounded function with \(\|m\|_\infty>0\), and \(c(x)\) is a continuous function which changes sign in \(\overline\Omega\).
Mathematics Subject Classification ID: 35J55
Mathematics Subject Classification ID: 35J50
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35B38
zbMATH DE Number: 5563048
zbMATH Keywords: nonlinear elliptic system
zbMATH Keywords: critical points
zbMATH Keywords: Nehari manifold
zbMATH Keywords: Dirichlet boundary condition
author: Ghasem A. Afrouzi
author: S. H. Rasouli
MaRDI profile type: Publication
full work available at URL: j.na.2008.10.118
OpenAlex ID: W2007476281
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DOI: 10.1016/J.NA.2008.10.118
description: scientific article; zbMATH DE number 5563048