We study the periodic boundary value problem associated with the second order nonlinear equation u''+(λa+(t)−μa−(t))g(u)=0, where g(u) has superlinear growth at zero and sublinear growth at infinity. For λ,μ positive and large, we prove the existence of 3^m−1 positive T-periodic solutions when the weight function a(t) has m positive humps separated by m negative ones (in a T-periodicity interval). As a byproduct of our approach we also provide abundance of positive subharmonic solutions and symbolic dynamics. The proof is based on coincidence degree theory for locally compact operators on open unbounded sets and also applies to Neumann and Dirichlet boundary conditions. Finally, we deal with radially symmetric positive solutions for the Neumann and the Dirichlet problems associated with elliptic PDEs.

%B Trans. Amer. Math. Soc. %I American Mathematical Society %G en %U http://urania.sissa.it/xmlui/handle/1963/35264 %1 35568 %2 Mathematics %4 1 %# MAT/05 %$ Approved for entry into archive by Maria Pia Calandra (calapia@sissa.it) on 2016-11-21T07:53:44Z (GMT) No. of bitstreams: 0 %0 Journal Article %J Communications in Contemporary Mathematics %D 2018 %T Positive subharmonic solutions to nonlinear ODEs with indefinite weight %A Alberto Boscaggin %A Guglielmo Feltrin %XWe prove that the superlinear indefinite equation u″ + a(t)up = 0, where p > 1 and a(t) is a T-periodic sign-changing function satisfying the (sharp) mean value condition ∫0Ta(t)dt < 0, has positive subharmonic solutions of order k for any large integer k, thus providing a further contribution to a problem raised by Butler in its pioneering paper [Rapid oscillation, nonextendability, and the existence of periodic solutions to second order nonlinear ordinary differential equations, J. Differential Equations 22 (1976) 467–477]. The proof, which applies to a larger class of indefinite equations, combines coincidence degree theory (yielding a positive harmonic solution) with the Poincaré–Birkhoff fixed point theorem (giving subharmonic solutions oscillating around it).

%B Communications in Contemporary Mathematics %V 20 %P 1750021 %G eng %U https://doi.org/10.1142/S0219199717500213 %R 10.1142/S0219199717500213 %0 Journal Article %J Proc. Roy. Soc. Edinburgh Sect. A 146 (2016), 449–474. %D 2016 %T Pairs of positive periodic solutions of nonlinear ODEs with indefinite weight: a topological degree approach for the super-sublinear case %A Alberto Boscaggin %A Guglielmo Feltrin %A Fabio Zanolin %XWe study the periodic and Neumann boundary value problems associated with the second order nonlinear differential equation u''+cu'+λa(t)g(u)=0, where g:[0,+∞[→[0,+∞[ is a sublinear function at infinity having superlinear growth at zero. We prove the existence of two positive solutions when ∫a(t)dt 0 is sufficiently large. Our approach is based on Mawhin's coincidence degree theory and index computations.

%B Proc. Roy. Soc. Edinburgh Sect. A 146 (2016), 449–474. %I Cambridge University Press %G en %U http://urania.sissa.it/xmlui/handle/1963/35262 %1 35566 %2 Mathematics %4 1 %# MAT/05 %$ Approved for entry into archive by Maria Pia Calandra (calapia@sissa.it) on 2016-11-21T07:33:12Z (GMT) No. of bitstreams: 0 %R 10.1017/S0308210515000621 %0 Journal Article %J Advanced Nonlinear Studies %D 2013 %T Pairs of nodal solutions for a class of nonlinear problems with one-sided growth conditions %A Alberto Boscaggin %A Fabio Zanolin %B Advanced Nonlinear Studies %I Advanced Nonlinear Studies, Inc. %V 13 %P 13–53 %G eng %R 10.1515/ans-2013-0103 %0 Journal Article %J Nonlinear Differential Equations and Applications NoDEA %D 2013 %T Planar Hamiltonian systems at resonance: the Ahmad–Lazer–Paul condition %A Alberto Boscaggin %A Maurizio Garrione %XWe consider the planar Hamiltonian system\$\$Ju^{\backslashprime} = \backslashnabla F(u) + \backslashnabla_u R(t,u), \backslashquad t \backslashin [0,T], \backslash,u \backslashin \backslashmathbb{R}^2,\$\$with F(u) positive and positively 2-homogeneous and \$\${\backslashnabla_{u}R(t, u)}\$\$sublinear in u. By means of an Ahmad-Lazer-Paul type condition, we prove the existence of a T-periodic solution when the system is at resonance. The proof exploits a symplectic change of coordinates which transforms the problem into a perturbation of a linear one. The relationship with the Landesman–Lazer condition is analyzed, as well.

%B Nonlinear Differential Equations and Applications NoDEA %V 20 %P 825–843 %8 Jun %G eng %U https://doi.org/10.1007/s00030-012-0181-2 %R 10.1007/s00030-012-0181-2 %0 Journal Article %J Discrete & Continuous Dynamical Systems - A %D 2013 %T Subharmonic solutions for nonlinear second order equations in presence of lower and upper solutions %A Alberto Boscaggin %A Fabio Zanolin %K lower and upper solutions %K parameter dependent equations %K Periodic solutions %K Poincaré-Birkhoff twist theorem %K subharmonic solutions %XWe study the problem of existence and multiplicity of subharmonic solutions for a second order nonlinear ODE in presence of lower and upper solutions. We show how such additional information can be used to obtain more precise multiplicity results. Applications are given to pendulum type equations and to Ambrosetti-Prodi results for parameter dependent equations.

%B Discrete & Continuous Dynamical Systems - A %V 33 %P 89 %G eng %U http://aimsciences.org//article/id/3638a93e-4f3e-4146-a927-3e8a64e6863f %R 10.3934/dcds.2013.33.89 %0 Journal Article %J Advanced Nonlinear Studies %D 2012 %T One-signed harmonic solutions and sign-changing subharmonic solutions to scalar second order differential equations %A Alberto Boscaggin %B Advanced Nonlinear Studies %I Advanced Nonlinear Studies, Inc. %V 12 %P 445–463 %G eng %R 10.1515/ans-2012-0302 %0 Journal Article %J Journal of Differential Equations %D 2012 %T Pairs of positive periodic solutions of second order nonlinear equations with indefinite weight %A Alberto Boscaggin %A Fabio Zanolin %K Critical points %K Necessary conditions %K Pairs of positive solutions %K Periodic solutions %XWe study the problem of the existence and multiplicity of positive periodic solutions to the scalar ODEu″+λa(t)g(u)=0,λ>0, where g(x) is a positive function on R+, superlinear at zero and sublinear at infinity, and a(t) is a T-periodic and sign indefinite weight with negative mean value. We first show the nonexistence of solutions for some classes of nonlinearities g(x) when λ is small. Then, using critical point theory, we prove the existence of at least two positive T-periodic solutions for λ large. Some examples are also provided.

%B Journal of Differential Equations %V 252 %P 2900 - 2921 %G eng %U http://www.sciencedirect.com/science/article/pii/S0022039611003895 %R https://doi.org/10.1016/j.jde.2011.09.011 %0 Journal Article %J Portugaliae Mathematica %D 2012 %T Periodic solutions to superlinear planar Hamiltonian systems %A Alberto Boscaggin %XWe prove the existence of infinitely many periodic (harmonic and subharmonic) solutions to planar Hamiltonian systems satisfying a suitable superlinearity condition at infinity. The proof relies on the Poincare-Birkhoff fixed point theorem.

%B Portugaliae Mathematica %I European Mathematical Society Publishing House %V 69 %P 127–141 %G eng %0 Journal Article %J Journal of Differential Equations %D 2012 %T Positive periodic solutions of second order nonlinear equations with indefinite weight: Multiplicity results and complex dynamics %A Alberto Boscaggin %A Fabio Zanolin %K Complex dynamics %K Poincaré map %K Positive periodic solutions %K Subharmonics %XWe prove the existence of a pair of positive T-periodic solutions as well as the existence of positive subharmonic solutions of any order and the presence of chaotic-like dynamics for the scalar second order ODEu″+aλ,μ(t)g(u)=0, where g(x) is a positive function on R+, superlinear at zero and sublinear at infinity, and aλ,μ(t) is a T-periodic and sign indefinite weight of the form λa+(t)−μa−(t), with λ,μ>0 and large.

%B Journal of Differential Equations %V 252 %P 2922 - 2950 %G eng %U http://www.sciencedirect.com/science/article/pii/S0022039611003883 %R https://doi.org/10.1016/j.jde.2011.09.010 %0 Journal Article %J Journal of Mathematical Analysis and Applications %D 2011 %T A note on a superlinear indefinite Neumann problem with multiple positive solutions %A Alberto Boscaggin %K Indefinite weight %K Nonlinear boundary value problems %K positive solutions %K Shooting method %XWe prove the existence of three positive solutions for the Neumann problem associated to u″+a(t)uγ+1=0, assuming that a(t) has two positive humps and ∫0Ta−(t)dt is large enough. Actually, the result holds true for a more general class of superlinear nonlinearities.

%B Journal of Mathematical Analysis and Applications %V 377 %P 259 - 268 %G eng %U http://www.sciencedirect.com/science/article/pii/S0022247X10008796 %R https://doi.org/10.1016/j.jmaa.2010.10.042 %0 Journal Article %J Nonlinear Analysis: Theory, Methods & Applications %D 2011 %T Resonance and rotation numbers for planar Hamiltonian systems: Multiplicity results via the Poincaré–Birkhoff theorem %A Alberto Boscaggin %A Maurizio Garrione %K Multiple periodic solutions %K Poincaré–Birkhoff theorem %K Resonance %K Rotation number %XIn the general setting of a planar first order system (0.1)u′=G(t,u),u∈R2, with G:[0,T]×R2→R2, we study the relationships between some classical nonresonance conditions (including the Landesman–Lazer one) — at infinity and, in the unforced case, i.e. G(t,0)≡0, at zero — and the rotation numbers of “large” and “small” solutions of (0.1), respectively. Such estimates are then used to establish, via the Poincaré–Birkhoff fixed point theorem, new multiplicity results for T-periodic solutions of unforced planar Hamiltonian systems Ju′=∇uH(t,u) and unforced undamped scalar second order equations x″+g(t,x)=0. In particular, by means of the Landesman–Lazer condition, we obtain sharp conclusions when the system is resonant at infinity.

%B Nonlinear Analysis: Theory, Methods & Applications %V 74 %P 4166 - 4185 %G eng %U http://www.sciencedirect.com/science/article/pii/S0362546X11001817 %R https://doi.org/10.1016/j.na.2011.03.051 %0 Journal Article %J Advanced Nonlinear Studies %D 2011 %T Subharmonic solutions of planar Hamiltonian systems: a rotation number approach %A Alberto Boscaggin %B Advanced Nonlinear Studies %I Advanced Nonlinear Studies, Inc. %V 11 %P 77–103 %G eng %R 10.1515/ans-2011-0104 %0 Journal Article %J Le Matematiche %D 2011 %T Subharmonic solutions of planar Hamiltonian systems via the Poincaré́-Birkhoff theorem %A Alberto Boscaggin %XWe revisit some recent results obtained in [1] about the existence of subharmonic solutions for a class of (nonautonomous) planar Hamiltonian systems, and we compare them with the existing literature. New applications to undamped second order equations are discussed, as well.

%B Le Matematiche %V 66 %P 115–122 %G eng