Nonlinear elastic beam problems with the parameter near resonance
type: Item
instance of: scholarly article
title: Nonlinear elastic beam problems with the parameter near resonance
zbMATH Open document ID: 1417.34054
author: Man Xu
author: Ruyun Ma
published in: Open Mathematics
publication date: 2019-03-29T00:00:00Z
review text: The article is devoted to the existence of solutions for the following fourth-order Dirichlet boundary value problem: \[ \begin{gathered} u^{(4)}(x)-\lambda u(x)=f(x,u(x))-h(x),\quad 0< x<1,\\ u(0)=u(1)=u'(0)=u'(1)=0, \end{gathered} \] where $\lambda$ is a parameter, $h\in C[0,1]$, and $f\in C([0,1]\times\mathbb{R},\mathbb{R})$. The nonlinearity $f$ is assumed to satisfy a weak monotonocity condition together with an asymptotic sublinear growth assumption. The authors first recall a spectrum result on the linear eigenvalue problem: \[ \begin{gathered} u^{(4)}(x)=\lambda u(x),\quad 0< x<1,\\ u(0)=u(1)=u'(0)=u'(1)=0, \end{gathered} \] a result which comes from their work [ibid. 16, 1176--1186 (2018; Zbl 07045499)]. Then the authors prove the existence of strict upper and lower solutions in case $h=0$. Finally The Rabinowitz global bifurcation technique at infinity is used to establish existence of connected component (second solution). An illustrating example with sublinear $f$ is supplied.
Mathematics Subject Classification ID: 34B09
Mathematics Subject Classification ID: 34B27
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 34A40
zbMATH DE Number: 7045499
zbMATH Keywords: elastic beam
zbMATH Keywords: at resonance
zbMATH Keywords: multiplicity
zbMATH Keywords: eigenvalue
zbMATH Keywords: lower and upper solutions
zbMATH Keywords: bifurcation
reviewed by: Smail Djebali
MaRDI profile type: Publication
full work available at URL: math-2018-0097
OpenAlex ID: W2899024472
cites work: Bifurcation from infinity and multiple solutions for some Dirichlet problems with unbounded nonlinearities
cites work: On some higher order boundary value problems at resonance
cites work: Nonlinear eigenvalue problems with the parameter near resonance
cites work: Unbounded perturbations of forced second order ordinary differential equations at resonance
cites work: Existence and multiplicity results for a class of nonlinear elliptic boundary value problems at resonance
cites work: Existence and multiplicity results for nonlinear elliptic problems with linear part at resonance. The case of the simple eigenvalue
cites work: Periodic solutions of asymptotically positively homogeneous differential equations
cites work: On an elliptic boundary value problem at double resonance
cites work: A Resonance Problem in Which the Nonlinearity May Grow Linearly
cites work: Bifurcation from infinity and multiple solutions for periodic boundary value problems
cites work: Existence and uniqueness results for the bending of an elastic beam equation at resonance
cites work: Infinitely many solutions of superlinear fourth order boundary value problems
cites work: Uniqueness and exact multiplicity of solutions for a class of fourth-order semilinear problems
cites work: \(L^ p\)-solvability of a fourth-order boundary-value problem at resonance
cites work: On solutions of a fourth-order Lidstone boundary value problem at resonance
cites work: On the lower and upper solutions method for the problem of elastic beam with hinged ends
cites work: Positivity and lower and upper solutions for fourth order boundary value problems
cites work: On positive solutions of some nonlinear fourth-order beam equations
cites work: The method of lower and upper solutions for fourth-order two-point boundary value problems
cites work: Q5703730
cites work: On bifurcation from infinity
cites work: Bifurcation of positive solutions of a nonlinear discrete fourth-order boundary value problem
cites work: Positive solutions of singular sublinear fourth-order boundary value problems
cites work: On a fourth-order boundary value problem at resonance
cites work: Q3999241
cites work: Spectrum of a linear fourth‐order differential operator and its applications
cites work: Global Analysis of Two-Parameter Elliptic Eigenvalue Problems
DOI: 10.1515/MATH-2018-0097
description: scientific article; zbMATH DE number 7045499