Variational and numerical aspects of a system of ODEs with concave-convex nonlinerities

TL;DR

This paper investigates a Hamiltonian system of ODEs with concave-convex nonlinearities, establishing solution multiplicity for certain parameters, analyzing solution properties, and proposing numerical methods to explore solution ranges.

Contribution

It introduces a numerical strategy using Poincaré-Miranda and shooting methods for systems of ODEs with concave-convex nonlinearities, complementing theoretical results.

Findings

Multiple nonnegative solutions exist for certain parameter ranges.

Solutions exhibit regularity and symmetry properties.

Numerical methods effectively explore solution multiplicity ranges.

Abstract

In this work we discuss a Hamiltonian system of ordinary differential equations under Dirichlet boundary conditions. The system of equations in consideration features a mixed (concave-convex) power nonlinearity depending on a positive parameter $\lambda$. We show multiplicity of nonnegative solutions of the system for a certain range of the parameter $\lambda$ and we also discuss regularity and symmetry of nonnegative solutions of the system. Besides, we present a numerical strategy aiming at the exploration of the optimal range of $\lambda$ for which multiplicity of solutions holds. The numerical experiments are based on the Poincar\'{e}-Miranda theorem and the shooting method, which have been lesser explored for systems of ODEs. Our work is motivated by the works of Ambrosetti et al., 1994 and Moreira dos Santos, 2009.