Nonlinear eigenvalue problem for a model equation of an elastic surface
type: Item
instance of: scholarly article
title: Nonlinear eigenvalue problem for a model equation of an elastic surface
zbMATH Open document ID: 0836.35112
author: Nobuyoshi Fukagai
author: Kimiaki Narukawa
published in: Hiroshima Mathematical Journal
publication date: 1996-04-29T00:00:00Z
review text: The existence of nonzero weak solutions of the boundary value problem \[ - \gamma \text{div} \left[ {\bigl( \sqrt {1 + |\nabla u |^2} - 1 \bigr)^{\gamma - 1} \over \sqrt {1 + |\nabla u |^2}} \nabla u \right] = \lambda f(x,u) \text{ in } \Omega \tag{1} \] \[ (2) \quad u \geq 0 \text{ in } \Omega \qquad (3) \quad u = 0 \text{ on } \partial \Omega \] is discussed for \(\gamma > 1\) and \(\lambda > 0\). In (1), \(\Omega\) is a bounded domain in \(\mathbb{R}^n\). For \(n = 1\) the derivation of (1) is given. In this case (1) describes the equilibrium configuration of a nonlinear elastic string and \(\lambda\) depends on the tension force in the string. The critical points of the functional \[ I_\lambda [u] = \int_\Omega \Bigl( \sqrt {1 + |\nabla u |^2} - 1 \Bigr)^\gamma dx - \lambda \int_{\Omega} F(x,u) dx \tag{4} \] where \[ F(x,u) = \int^u_0 f(x, \xi) d \xi \tag{5} \] are the solutions of (1), (3). Under some conditions on \(F(x,u)\) the existence of the local minimizer of \(I_\lambda [u]\) is proven. The second critical point of \(I_\lambda [u]\) is obtained and the asymptotic behavior of the secondary solution is analysed. The special case, when \(f(x,u) = qu^{q - 1}\), is investigated in detail.
Mathematics Subject Classification ID: 35P30
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35Q72
Mathematics Subject Classification ID: 74K05
Mathematics Subject Classification ID: 53A05
Mathematics Subject Classification ID: 35D05
zbMATH DE Number: 774753
zbMATH Keywords: critical point
zbMATH Keywords: elastic surface
zbMATH Keywords: nonlinear elastic string
zbMATH Keywords: local minimizer
zbMATH Keywords: asymptotic behavior
reviewed by: István Ecsedi
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.32917/hmj/1206127823
OpenAlex ID: W2107704020
Wikidata QID: Q128899398
DOI: 10.32917/HMJ/1206127823
description: scientific article; zbMATH DE number 774753