Some bifurcation results for rapidly growing nonlinearities
type: Item
instance of: scholarly article
title: Some bifurcation results for rapidly growing nonlinearities
zbMATH Open document ID: 1263.35026
published in: Discrete and Continuous Dynamical Systems
publication date: 2013-03-19T00:00:00Z
review text: Let \(\Omega\) be a bounded smooth domain in \({\mathbb R}^N\). This paper deals with the quasilinear elliptic problem \[ -\Delta u=\lambda f(u)\qquad\text{in}\;\Omega, \] under the Dirichlet boundary condition \(u=0\) on \(\partial\Omega\). In the first part it is assumed that \(f(t)\) behaves like \(\exp t^\alpha\) as \(t\rightarrow\infty\), for some \(\alpha>0\). The main result in this case establishes that the upper limit of the dimension \(N\) is independent of \(\alpha\). This is a surprising difference from variational methods where the size of \(\alpha\) often seems important. Next, it is studied the case where \(f(t)=(1-\varepsilon t)^k\exp t\), with \(\varepsilon>0\). Such nonlinearities frequently occur in both the theory of catalysts and combustion theory. The author proves that there is a connected branch of positive solutions with many bifurcations. The method developed in the present paper works for many other small perturbations of \(\exp t\), even many nonlocal ones. In the final part it is established the existence of an unbounded closed connected subset of the set of solutions which has infinitely many solutions and these are unbounded.
reviewed by: Vicenţiu D. Rădulescu
Mathematics Subject Classification ID: 35B32
Mathematics Subject Classification ID: 35A15
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 35J61
zbMATH DE Number: 6146679
zbMATH Keywords: nonlinear elliptic equation
zbMATH Keywords: rapidly growing nonlinearities
zbMATH Keywords: bifurcation
zbMATH Keywords: many bifurcations
zbMATH Keywords: infinitely many solutions
author: E. N. Dancer
MaRDI profile type: Publication
full work available at URL: dcds.2013.33.153
OpenAlex ID: W2320765168
DOI: 10.3934/DCDS.2013.33.153
description: scientific article; zbMATH DE number 6146679