On some fourth-order semilinear elliptic problems in \(\mathbb R^N\)
type: Item
instance of: scholarly article
title: On some fourth-order semilinear elliptic problems in \(\mathbb R^N\)
zbMATH Open document ID: 1011.35045
DOI: 10.1016/S0362-546X(01)00144-4
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2002-06-27T00:00:00Z
review text: The purpose of this paper is to establish the existence of at least two solutions \(u\in D^{2,2}(\mathbb R^N)\setminus \{ 0\}\) to the problem \(\Delta^2u-\lambda g(x)u=f(x)|u|^{p-2}u\) in \(\mathbb R^N\), where \(N\geq 5\), \(\lambda >0\), \(p=2N/(N-4)\), and \(f\) changes sign in a prescribed manner. The proof of the main result is based on variational tools and it can be summarized as follows. The authors first examine Palais-Smale sequences in order to find the range of a variational functional associated to the above problem for which the Palais-Smale condition holds. Then it is applied the mountain-pass principle in order to obtain a first solution of the problem. Next, it is established the existence of a second solution, provided \(\lambda\) is in a small right neighbourhood of the first eigenvalue to the corresponding weighted eigenvalue problem. The key fact in the proof of the last assertion is that \(f(x)\) changes sign. The paper gives an interesting perspective for the treatment of wide classes of fourth-order semilinear problems with lack of compactness.
Mathematics Subject Classification ID: 35J30
Mathematics Subject Classification ID: 31B30
Mathematics Subject Classification ID: 49J35
Mathematics Subject Classification ID: 58E05
zbMATH DE Number: 1761024
zbMATH Keywords: biharmonic operator
zbMATH Keywords: principal eigenvalue
zbMATH Keywords: constrained minimization
zbMATH Keywords: critical growth
zbMATH Keywords: variational methods
zbMATH Keywords: Palais-Smale condition
zbMATH Keywords: mountain-pass principle
author: Jan Chabrowski
author: João Marcos do Ó
reviewed by: Vicenţiu D. Rădulescu
MaRDI profile type: Publication
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full work available at URL: s0362-546x(01)00144-4
OpenAlex ID: W1998135078
description: scientific article; zbMATH DE number 1761024