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

Abstract

We study a Hamiltonian system of ordinary differential equations under Dirichlet boundary conditions. The system features a mixed (concave-convex) power nonlinearity with a positive parameter. We show multiplicity of nonnegative solutions for a range of the parameter and discuss the regularity and symmetry of nonnegative solutions. Besides, we present a numerical strategy aiming at the exploration of the optimal range of for which multiplicity of positive solutions holds. The numerical experiments are based on the Poincare-Miranda Theorem and the shooting method, which have been lesser explored in the context of multiple positive solutions of systems of ODEs. Our work has been motivated by the results in Ambrosetti et al. in [4] and Moreira dos Santos in [18].