On nontrivial solutions of an asymptotically linear Dirichlet problem
type: Item
instance of: scholarly article
title: On nontrivial solutions of an asymptotically linear Dirichlet problem
zbMATH Open document ID: 0667.35025
DOI: 10.1016/0022-0396(88)90131-3
author: Nguyên Phuong Các
published in: Journal of Differential Equations
publication date: 1988-01-01T00:00:00Z
review text: The author adds to the large existing literature dealing with solutions of \[ -\Delta u=f(u)\quad in\quad \Omega,\quad f(0)=0;\quad u\equiv 0\quad on\quad \partial \Omega, \] where \(\Omega\) is a bounded domain in \({\mathbb{R}}^ n\), and \(\partial \Omega\) is smooth. He assumes \(\lim_{\zeta \to \pm \infty} f(\zeta)/\zeta\) exists. However, he does not require differentiability of f(\(\zeta)\), and the interval \([f_-,f_+]\) is allowed to contain an eigenvalue of - \(\Delta\). He proves that if there exists a constant \(\beta\), such that \[ (f(\zeta)-f(\eta))/(\zeta -\eta)\leq \beta <\lambda_{k+1},\quad (k+1\quad eigenvalue) \] \(\forall \zeta\), \(\eta\in {\mathbb{R}}\), \(\zeta\neq \eta\), and \(\exists \delta >0\), such that \(0<| \zeta | <\delta \Rightarrow f(\zeta)/\zeta <\lambda_ k\) then the Dirichlet problem stated above has a solution for negative values of Galluët-Kavakian numbers. Other, related results are also proved.
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 35A05
zbMATH DE Number: 4091985
zbMATH Keywords: differentiability
zbMATH Keywords: eigenvalue
zbMATH Keywords: Dirichlet problem
reviewed by: Vadim Komkov
MaRDI profile type: Publication
full work available at URL: 0022-0396(88)90131-3
OpenAlex ID: W2054658516
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description: scientific article; zbMATH DE number 4091985