Existence and multiplicity of solutions to nonlinear differential equations at resonance
type: Item
instance of: scholarly article
title: Existence and multiplicity of solutions to nonlinear differential equations at resonance
zbMATH Open document ID: 0531.35037
DOI: 10.1016/0022-247X(83)90015-X
author: R. Kent Nagle
author: Karen Singkofer
published in: Journal of Mathematical Analysis and Applications
publication date: 1983-01-01T00:00:00Z
review text: The problem \(Lu+g(u)=f\) in \(\Omega\), \(B_ ku=0\), \(k=1,...,m-1\), where L is a uniformly elliptic differential operator of order 2m, \(B_ k\) are boundary operators, \(\Omega\) is a smooth domain, and g together with its derivative g' are continuous and bounded, is considered in this paper. It is assumed that the kernel of L is a one-dimensional subspace spanned by a function \(\theta\) (x) satisfying \(0<\alpha \leq \theta(x)\leq \beta\), with \(\alpha\) and \(\beta\) constants, a very strong assumption. Existence and multiplicity of solutions for small f is studied under additional assumptions, concerning mainly the zeroes of g. The alternative method of Cesari is the main tool used by the authors. The results are, in some sense, complementary of the previous ones where Landesman-Lazer type assumptions are made by several authors (Fučik, Hess, Ambrosetti-Mancini, etc.). Section 2 includes some fundamental estimates on the solutions. Existence results are proved in Section 3 for g with a finite (or countable) number of separated zeroes, and multiplicity results are given in Section 4.
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35J40
Mathematics Subject Classification ID: 35B45
Mathematics Subject Classification ID: 35A05
zbMATH DE Number: 3842366
zbMATH Keywords: existence
zbMATH Keywords: multiplicity
zbMATH Keywords: resonance
zbMATH Keywords: alternative method
zbMATH Keywords: Landesman-Lazer condition
reviewed by: Jesús Hernández
MaRDI profile type: Publication
cites work: On kernels, eigenvalues, and eigenfunctions of operators related to elliptic problems
cites work: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I
cites work: Q4155227
cites work: On the inversion of some differentiable mappings with singularities between Banach spaces
cites work: Q4167193
cites work: Q5341132
cites work: Q4112515
cites work: Q4768228
cites work: Q4059290
cites work: Boundary value problems with jumping nonlinearities
cites work: Q4200572
cites work: Boundary value problems with bounded nonlinearity and general null-space of the linear part
cites work: Q4173739
cites work: Q5588800
cites work: Strongly nonlinear perturbations of nonnegative boundary value problems with kernel
cites work: Equations with unbounded nonlinearities
cites work: Nonlinear ordinary differential equations at resonance with slowly varying nonlinearities
cites work: Alternative problems and monotonicity
cites work: Q5533735
cites work: A nonlinear elliptic boundary value problem at resonance
cites work: A sharp sufficient condition for solution of a nonlinear elliptic boundary value problem
full work available at URL: 0022-247x(83)90015-x
OpenAlex ID: W2009330103
description: scientific article; zbMATH DE number 3842366