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

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.