Multiplicity of solutions for a fourth order problem with exponential nonlinearity
type: Item
instance of: scholarly article
title: Multiplicity of solutions for a fourth order problem with exponential nonlinearity
zbMATH Open document ID: 1190.34017
author: Juan Dávila
author: Isabel Flores
author: Ignacio Guerra
published in: Journal of Differential Equations
publication date: 2009-11-23T00:00:00Z
review text: Consider for \(\lambda \geq 0,\) the boundary value problem \[ \Delta^2 u = \lambda e^u\text{ in the unit ball }B, \] \[ u=0, \partial u /\partial n =0\text{ on }\partial B \] (\(n\) is the exterior unit normal vector). First, it is proved that if \(N \geq 5\), there exists a unique \(\lambda_S>0\) such that the given problem with \(\lambda=\lambda_S\) admits a radial singular solution; moreover, this solution is unique in the class of radial solutions. In order to state the main result, one has to consider the largest value of \(\lambda \geq 0\), denoted by \(\lambda^*,\) such that the given problem has a radial classical solution. It is then proved that if \(5 \leq N \leq 12\) then \(\lambda_S < \lambda^*\) and with \(\lambda = \lambda^*\) there exist infinitely many regular radial solutions. For \(\lambda \neq \lambda_S\) there exist a finite number of regular radial solutions, and this number goes to infinity as \(\lambda \to \lambda_S\). Finally, in the case \(N\geq 13\) it is proved that \(\lambda_S = \lambda^*\) and for all \(0 < \lambda < \lambda^*\) there exists a unique regular radial solution. For \(\lambda = \lambda^*\) there exists a unique singular radial solution. Variants of the above results are performed in case of Navier boundary conditions. For the proofs of the multiplicity results, the ODE arising from the given problem when searching radial solutions is trasformed into a suitable first-order 4-dimensional autonomous system; then phase space analysis methods are used.
reviewed by: Anna Capietto
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 35J40
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 34B09
zbMATH DE Number: 5637048
zbMATH Keywords: fourth-order elliptic problem
zbMATH Keywords: radial solutions
zbMATH Keywords: multiplicity
zbMATH Keywords: phase space analysis
MaRDI profile type: Publication
full work available at URL: j.jde.2009.07.023
OpenAlex ID: W2007251457
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DOI: 10.1016/J.JDE.2009.07.023
description: scientific article; zbMATH DE number 5637048