Electron. J. Differential Equations, Vol. 2025 (2025), No. 92, pp. 1-19.
Variational and numerical aspects of a system of ODEs with concave-convex nonlinearities
Oscar Agudelo, Gabriela Holubova, Martin Kudlac
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 \(\lambda\). We show multiplicity of nonnegative solutions
for a range of the parameter \(\lambda\) and discuss the regularity and symmetry of
nonnegative solutions. Besides, we present a numerical strategy aiming at the
exploration of the optimal range of \(\lambda\) 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].
Submitted January 12, 2025. Published October 3, 2025.
Math Subject Classifications: 34A34, 34B08, 34B18, 35J35.
Key Words: Hamiltonian system of odes; concave and convex nonlinearities; minimization theorem; mountain pass theorem; shooting method; moving planes method.
DOI: 10.58997/ejde.2025.92
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Oscar Agudelo
Department of Mathematics and NTIS
Faculty of Applied Sciences
University of West Bohemia in Pilsen
Univerzitní 8, 301 00 Plzen, Czech Republic
email: oiagudel@kma.zcu.cz, orcid: 0000-0002-2588-9999
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Gabriela Holubová
Department of Mathematics and NTIS
Faculty of Applied Sciences
University of West Bohemia in Pilsen
Univerzitní 8, 301 00 Plzen, Czech Republic
email: gabriela@kma.zcu.cz, orcid: 0000-0003-1127-3381
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Martin Kudlác
Department of Mathematiccfon; and NTIS
Faculty of Applied Sciences
University of West Bohemia in Pilsen
Univerzitní 8, 301 00 Plzen, Czech Republic
email: kudlacm@kma.zcu.cz, orcid: 009-0007-1749-3314
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