Multiplicity results for systems of superlinear second order equations
type: Item
instance of: scholarly article
title: Multiplicity results for systems of superlinear second order equations
zbMATH Open document ID: 0972.34012
author: Anna Capietto
author: Walter Dambrosio
published in: Journal of Mathematical Analysis and Applications
publication date: 2001-11-04T00:00:00Z
review text: The authors study the existence of solutions to the Dirichlet problem \[ x''+\nabla F(t,x)=q(t,x,x'),\quad x(0)=0,\quad x(\pi)=0, \] where \(F:[0,\pi]\times \mathbb R^m\to\mathbb R\) is a \({\mathcal C}^1\)-function and \(q:[0,\pi]\times \mathbb R^{2m}\to\mathbb R^m\) is continuous. The main result proves that for any \((n_1,\ldots,n_m)\in{\mathbb N}^m\) there exist \(2^m\) solutions so that the \(i\)th component of each of them has exactly \(n_i\) zeros. These are large solutions which are obtained assuming a superlinear behavior of the nonlinearity at infinity \[ \lim_{|x_i|\to\infty}\frac{1}{x_i} \frac{\partial F}{\partial x_i}(t,x)= +\infty,\quad |q_i(t,x,y)|\leq A_1|x_i|+ A_2|y_i|+ A_3, \] together with \[ x_i\frac{\partial F}{\partial x_i}(t,x)\geq K_0\quad\text{and}\quad\left |\frac{\partial F}{\partial t}(t,x)\right |\leq \alpha(t)F(t,x)+C. \] As corollaries, the authors obtain results on the Liénard equation and on weakly coupled systems. The proof uses a topological degree approach. The system is homotoped to uncoupled equations to which a time-map technique can be applied. An important step is based on the ``elastic property'', i.e. a control of the maximum of the norm of a solution in phase planes from an estimate on its minimum.
Mathematics Subject Classification ID: 34B15
zbMATH DE Number: 1523554
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: systems of second-order equations
zbMATH Keywords: superlinear problem
zbMATH Keywords: weakly coupled systems
zbMATH Keywords: Liénard equation
zbMATH Keywords: existence of solutions
zbMATH Keywords: multiplicity
zbMATH Keywords: nodal properties
zbMATH Keywords: degree theory
reviewed by: Patrick Habets
MaRDI profile type: Publication
full work available at URL: jmaa.2000.6942
OpenAlex ID: W2057517198
cites work: Forced vibrations of superquadratic Hamiltonian systems
cites work: Existence of forced oscillations for some nonlinear differential equations
cites work: Q4525518
cites work: A topological degree approach to sublinear systems of second order differential equations.
cites work: A continuation approach to some forced superlinear Sturm-Liouville boundary value problems
cites work: Q3947949
cites work: On the Existence of Radial Solutions of a Nonlinear Elliptic BVP in an Annulus
cites work: On the existence of many solutions for a class of superlinear elliptic systems
cites work: Multiple solutions of weakly-coupled systems with \(p\)-Laplacian operators
cites work: Periodic solutions of second order differential equations with superlinear asymmetric nonlinearities
cites work: Strongly nonlinear second-order ODEs with rapidly growing terms
cites work: Infinitely many solutions for a Dirichlet problem with a nonhomogeneous \(p\)-Laplacian-like operator in a ball
cites work: Infinitely many solutions of weakly coupled superlinear systems
cites work: Periodic solution for nonlinear systems with \(p\)-Laplacian-like operators
cites work: Q3049152
cites work: Critical point theory and Hamiltonian systems
cites work: On subharmonic solutions of hamiltonian systems
cites work: Multiple Critical Points of Perturbed Symmetric Functionals
cites work: Q3831323
cites work: Multiple solutions of anticoercive boundary value problems for a class of ordinary differential equations of second order
cites work: Solutions of prescribed number of zeroes to a class of superlinear ODE's systems
DOI: 10.1006/JMAA.2000.6942
Recommended article: MULTIPLICITY RESULTS FOR A CLASS OF SECOND ORDER SUPERLINEAR DIFFERENCE SYSTEMS
Recommended article: Multiplicity results for systems of asymptotically linear second order equations
Recommended article: Q3511705
Recommended article: Multiplicity results for second-order two-point boundary value problems with superlinear or sublinear nonlinearities
Recommended article: Q3728470
Recommended article: Existence and multiplicity of solutions for a class of superlinear elliptic systems
Recommended article: Multiplicity results for a class of superlinear elliptic problems
Recommended article: Multiplicity of solutions for elliptic system involving supercritical Sobolev exponent
Recommended article: Multiplicity of solutions for a superlinear \(p\)-Laplacian equation
Recommended article: On the existence of multiple positive solutions to some superlinear systems
description: scientific article; zbMATH DE number 1523554