On existence and stability of solutions for higher order semilinear Dirichlet problems
type: Item
instance of: scholarly article
title: On existence and stability of solutions for higher order semilinear Dirichlet problems
zbMATH Open document ID: 1201.34035
author: Marek Galewski
published in: Proceedings of the Indian Academy of Sciences. Mathematical Sciences
publication date: 2009-10-15T00:00:00Z
full work available at URL: http://hdl.handle.net/10338.dmlcz/128195
review text: The author considers the fourth order ordinary differential equation \[ \beta_kx^{iv}+\gamma_kx''+\delta_kx = F^k_x(t,x) \] with zero Dirichlet boundary data, where \(\beta_k>0\), \(F^k_x(t,0)\neq 0\) and \(x\mapsto F^k(t,x)\) is convex for \(x\) in a certain interval \([-d_0,d_0]\). It is shown that, under appropriate conditions, this equation has a solution \(x_k\). The proof is achieved by minimization of a dual functional obtained by means of the Fenchel conjugate of \(F^k\). The following stability result is also shown. Suppose \(k=1,2,\ldots\) and \(\beta_k,\gamma_k,\delta_k,F^k\) converge as \(k\to\infty\) (\(\beta_k>0\) decreasing). Then, passing to a subsequence, \(x_k\to\bar x\) and \(\bar x\) is a solution of a limiting problem. If \(\beta_k\to \beta_0>0\), the limiting problem is of fourth order, and if \(\beta_k\to 0\), it is of second order.
reviewed by: Andrzej Szulkin
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 47N10
Mathematics Subject Classification ID: 47J30
zbMATH DE Number: 5615377
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: dual variational method
zbMATH Keywords: existence of solutions
zbMATH Keywords: stability
MaRDI profile type: Publication
OpenAlex ID: W2017830123
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DOI: 10.1007/S12044-008-0048-Y
description: scientific article; zbMATH DE number 5615377