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.

PB - American Mathematical Society UR - http://urania.sissa.it/xmlui/handle/1963/35264 N1 - AMS Subject Classification: 34B15, 34B18, 34C25, 34C28, 47H11. U1 - 35568 U2 - Mathematics U4 - 1 U5 - MAT/05 ER - TY - JOUR T1 - Positive subharmonic solutions to nonlinear ODEs with indefinite weight JF - Communications in Contemporary Mathematics Y1 - 2018 A1 - Alberto Boscaggin A1 - Guglielmo Feltrin AB -We 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).

VL - 20 UR - https://doi.org/10.1142/S0219199717500213 ER - TY - JOUR T1 - Pairs of positive periodic solutions of nonlinear ODEs with indefinite weight: a topological degree approach for the super-sublinear case JF - Proc. Roy. Soc. Edinburgh Sect. A 146 (2016), 449–474. Y1 - 2016 A1 - Alberto Boscaggin A1 - Guglielmo Feltrin A1 - Fabio Zanolin AB -We 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.

PB - Cambridge University Press UR - http://urania.sissa.it/xmlui/handle/1963/35262 N1 - AMS Subject Classification: Primary 34B18; 34C25; Secondary 34B15; 47H11; U1 - 35566 U2 - Mathematics U4 - 1 U5 - MAT/05 ER - TY - JOUR T1 - Pairs of nodal solutions for a class of nonlinear problems with one-sided growth conditions JF - Advanced Nonlinear Studies Y1 - 2013 A1 - Alberto Boscaggin A1 - Fabio Zanolin PB - Advanced Nonlinear Studies, Inc. VL - 13 ER - TY - JOUR T1 - Planar Hamiltonian systems at resonance: the Ahmad–Lazer–Paul condition JF - Nonlinear Differential Equations and Applications NoDEA Y1 - 2013 A1 - Alberto Boscaggin A1 - Maurizio Garrione AB -We 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.

VL - 20 UR - https://doi.org/10.1007/s00030-012-0181-2 ER - TY - JOUR T1 - Subharmonic solutions for nonlinear second order equations in presence of lower and upper solutions JF - Discrete & Continuous Dynamical Systems - A Y1 - 2013 A1 - Alberto Boscaggin A1 - Fabio Zanolin KW - lower and upper solutions KW - parameter dependent equations KW - Periodic solutions KW - Poincaré-Birkhoff twist theorem KW - subharmonic solutions AB -We 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.

VL - 33 UR - http://aimsciences.org//article/id/3638a93e-4f3e-4146-a927-3e8a64e6863f ER - TY - JOUR T1 - One-signed harmonic solutions and sign-changing subharmonic solutions to scalar second order differential equations JF - Advanced Nonlinear Studies Y1 - 2012 A1 - Alberto Boscaggin PB - Advanced Nonlinear Studies, Inc. VL - 12 ER - TY - JOUR T1 - Pairs of positive periodic solutions of second order nonlinear equations with indefinite weight JF - Journal of Differential Equations Y1 - 2012 A1 - Alberto Boscaggin A1 - Fabio Zanolin KW - Critical points KW - Necessary conditions KW - Pairs of positive solutions KW - Periodic solutions AB -We 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.

VL - 252 UR - http://www.sciencedirect.com/science/article/pii/S0022039611003895 ER - TY - JOUR T1 - Periodic solutions to superlinear planar Hamiltonian systems JF - Portugaliae Mathematica Y1 - 2012 A1 - Alberto Boscaggin AB -We 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.

PB - European Mathematical Society Publishing House VL - 69 ER - TY - JOUR T1 - Positive periodic solutions of second order nonlinear equations with indefinite weight: Multiplicity results and complex dynamics JF - Journal of Differential Equations Y1 - 2012 A1 - Alberto Boscaggin A1 - Fabio Zanolin KW - Complex dynamics KW - Poincaré map KW - Positive periodic solutions KW - Subharmonics AB -We 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.

VL - 252 UR - http://www.sciencedirect.com/science/article/pii/S0022039611003883 ER - TY - JOUR T1 - A note on a superlinear indefinite Neumann problem with multiple positive solutions JF - Journal of Mathematical Analysis and Applications Y1 - 2011 A1 - Alberto Boscaggin KW - Indefinite weight KW - Nonlinear boundary value problems KW - positive solutions KW - Shooting method AB -We 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.

VL - 377 UR - http://www.sciencedirect.com/science/article/pii/S0022247X10008796 ER - TY - JOUR T1 - Resonance and rotation numbers for planar Hamiltonian systems: Multiplicity results via the Poincaré–Birkhoff theorem JF - Nonlinear Analysis: Theory, Methods & Applications Y1 - 2011 A1 - Alberto Boscaggin A1 - Maurizio Garrione KW - Multiple periodic solutions KW - Poincaré–Birkhoff theorem KW - Resonance KW - Rotation number AB -In 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.

VL - 74 UR - http://www.sciencedirect.com/science/article/pii/S0362546X11001817 ER - TY - JOUR T1 - Subharmonic solutions of planar Hamiltonian systems: a rotation number approach JF - Advanced Nonlinear Studies Y1 - 2011 A1 - Alberto Boscaggin PB - Advanced Nonlinear Studies, Inc. VL - 11 ER - TY - JOUR T1 - Subharmonic solutions of planar Hamiltonian systems via the Poincaré́-Birkhoff theorem JF - Le Matematiche Y1 - 2011 A1 - Alberto Boscaggin AB -We 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.

VL - 66 ER -