Some existence theorems for the sublinear Neumann boundary value problem
type: Item
instance of: scholarly article
title: Some existence theorems for the sublinear Neumann boundary value problem
zbMATH Open document ID: 1033.35029
DOI: 10.1016/S0362-546X(00)00230-3
author: Chun-Lei Tang
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2004-03-08T00:00:00Z
review text: The author studies the Neumann boundary value problem in the cases of both \(k= 1\) and \(k> 1\): \[ -\Delta u= \mu_k u+ g(x,u)\quad\text{in }\Omega,\quad {\partial u\over\partial n}= 0\quad\text{on }\partial\Omega,\tag{1} \] where \(\Omega\subset \mathbb{R}^N\) \((N> 1)\) is a bounded domain with smooth boundary and outward normal \(n(x)\) and \({\partial u\over\partial n}= n(x)\cdot\nabla u\), \(\mu_k\) is the \(k\)th distinct eigenvalue of the eigenvalue problem \[ -\Delta u= \mu u\quad\text{in }\Omega,\quad {\partial u\over\partial n}= 0\quad\text{on }\partial\Omega, \] \(\mu_1= 0\), \(g:\overline\Omega\times \mathbb{R}\to\mathbb{R}\) is a Carathéodory function. Under some assumptions on the function \(g(x,t)\), using the minimax method in critical point theory and saddle theorem, the author obtains some existence theorems for the solution of problem (1), i.e. problem (1) has at least one solution in the Sobolev space \(H^1(\Omega)\).
reviewed by: Zeng Yuesheng
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 35J65
zbMATH DE Number: 1741841
zbMATH Keywords: Neumann problem
zbMATH Keywords: resonance
zbMATH Keywords: saddle theorem
zbMATH Keywords: critical point
zbMATH Keywords: Sobolev embedding theorem
zbMATH Keywords: Poincaré inequality
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1741841