Global bifurcation results for a semilinear biharmonic equation on all of \(\mathbb{R}^N\)
type: Item
instance of: scholarly article
title: Global bifurcation results for a semilinear biharmonic equation on all of \(\mathbb{R}^N\)
zbMATH Open document ID: 0933.35063
DOI: 10.4171/ZAA/910
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 1999-12-20T00:00:00Z
full work available at URL: https://eudml.org/doc/48359
review text: Summary: We prove existence of positive solutions for the semilinear problem \[ (-\Delta)^2u=\lambda g(x)f(u),\quad u(x)>0\;(x\in\mathbb{R}^N), \quad\lim_{ |x|\to +\infty} u(x)=0 \] under the main hypothesis \(N>4\) and \(g\in L^{N/4}(\mathbb{R}^N)\). First, we employ classical spectral analysis for the existence of a simple positive principal eigenvalue for the linearized problem. Next, we prove the existence of a global continuum of positive solutions for the problem above, branching out from the first eigenvalue of the differential equation in the case that \(f(u)=u\). This fact is achieved by applying standard local and global bifurcation theory. It was possible to carry out these methods by working between certain equivalent weighted and homogeneous Sobolev spaces.
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35B32
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35J70
Mathematics Subject Classification ID: 35B50
zbMATH DE Number: 1356407
zbMATH Keywords: nonlinear eigenvalue problems
zbMATH Keywords: local and global bifurcation theory
zbMATH Keywords: maximum principle
zbMATH Keywords: indefinite weights
author: Nikos M. Stavrakakis
author: Nikolaos B. Zographopoulos
MaRDI profile type: Publication
cites work: Q3269389
cites work: Decaying Solutions Of 2mth Order Elliptic Problems
cites work: Elliptic and parabolic semilinear problems without conditions at infinity
cites work: Global bifurcation results for a semilinear elliptic equation on all of \(\mathbb{R}^ N\)
cites work: On the construction of super and subsolutions for elliptic equations on all of N
cites work: Positive splutions of semilinear elliptic systems
cites work: Bifurcation from simple eigenvalues
cites work: Elliptic Equations of Order 2m in Annular Domains
cites work: Critical exponents, critical dimensions and the biharmonic operator
cites work: Eigenvalues of Elliptic Boundary Value Problems With an Indefinite Weight Function
cites work: Principal eigenvalues for some quasilinear elliptic equations on \(\mathbb R^N\)
cites work: Symmetry and related properties via the maximum principle
cites work: Q5187597
cites work: Q4018487
cites work: Radial entire solutions to even order semilinear elliptic equations in the plane
cites work: Radial Entire Solutions of Even Order Semilinear Elliptic Equations
cites work: Existence and regularity of weak solutions to the Dirichletproblem for semilinear elliptic systems of higher order.
cites work: A rellich type identity and applications
cites work: Critical semilinear biharmonic equations in RN
cites work: Existence and non-existence of positive solutions of nonlinear elliptic systems and the biharmonic equation
cites work: Q4888372
cites work: Critical exponents and critical dimensions of polyharmonic operators
cites work: Some global results for nonlinear eigenvalue problems
cites work: Q4375148
cites work: Q4363074
cites work: Positivity for a noncooperative system of elliptic equations in \({\mathbb{R}}^N\)
cites work: The best sobolev constant
cites work: Uniqueness for semilinear polyharmonic problems
cites work: Radial polyharmonic problems in \(\mathbb{R}^ N\)
cites work: Q3995594
OpenAlex ID: W1967124761
description: scientific article; zbMATH DE number 1356407