Multiplicity of solutions of semilinear elliptic boundary value problems with jumping nonlinearities at zero
type: Item
instance of: scholarly article
title: Multiplicity of solutions of semilinear elliptic boundary value problems with jumping nonlinearities at zero
zbMATH Open document ID: 1033.35038
DOI: 10.1016/S0362-546X(00)00234-0
author: Zhaoli Liu
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2004-03-08T00:00:00Z
review text: This paper deals with the elliptic boundary value problem \[ -\Delta u= f(u),\quad x\in\Omega;\quad u=0,\quad x\in\partial\Omega,\tag{1} \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\) with smooth boundary \(\partial\Omega\) and \(f: \mathbb{R}\to\mathbb{R}\) is a Lipschitz continuous function. By using the \(L^p\) theory of elliptic operators and Sobolev imbedding theorems, the mountain pass lemma and fixed point index, the author proves that problem (1) has a minimal positive solution \(u_1\), a maximal negative solution \(u_2\) and a sign-changing solution \(u_3\), \(u_2< u_3< u_1\), under some conditions. Furthermore, under some additional assumptions, the author proves that problem (1) has at least nontrivial solutions \(u_1\), \(u_2\), \(u_3\), \(u_4\), and \(u_1\) is the minimal positive solution, \(u_2\) is the maximal negative solution, both \(u_3\) and \(u_4\) are sign-changing solutions, \(u_2< u_3< u_1\), \(u_2< u_4< u_1\).
reviewed by: Zeng Yuesheng
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35J20
zbMATH DE Number: 1741845
zbMATH Keywords: sub- and super-solutions
zbMATH Keywords: Sobolev imbedding theorems
zbMATH Keywords: mountain pass lemma
zbMATH Keywords: fixed point index
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1741845