A fourth order elliptic equation with nonlinear boundary conditions
type: Item
instance of: scholarly article
title: A fourth order elliptic equation with nonlinear boundary conditions
zbMATH Open document ID: 1014.35021
DOI: 10.1016/S0362-546X(01)00718-0
author: Julián Fernández Bonder
author: J. D. Rossi
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2002-08-25T00:00:00Z
review text: The authors deal with the existence of nontrivial solutions of the following fourth-order problem: \[ \begin{cases} -\Delta^2u= u\quad & \text{in } \Omega\\ -{\partial\Delta u\over\partial n}= f(x,u)\quad & \text{on }\partial \Omega,\\ \Delta u=0 \text{ or }{\partial u\over\partial n}= 0\quad & \text{on }\partial\Omega,\end{cases} \tag{1} \] where \(\Omega\) is a bounded smooth domain in \(\mathbb R^N\), \({\partial\over \partial n}\) is the outer normal derivative and \(f: \partial \Omega\times \mathbb R\to\mathbb R\) is a given function. Under suitable assumptions on \(f(x,u)\) the authors prove the existence of infinitely many weak solutions for (1). To this end, they use abstract results from critical point theory, some topological tools and Sobolev trace inequalities.
Mathematics Subject Classification ID: 35J40
Mathematics Subject Classification ID: 35J65
zbMATH DE Number: 1787982
zbMATH Keywords: trace inequality
zbMATH Keywords: critical point theory
zbMATH Keywords: topological tools
reviewed by: Messoud Efendiev
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1787982