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.

1 aBoscaggin, Alberto1 aFeltrin, Guglielmo1 aZanolin, Fabio uhttp://urania.sissa.it/xmlui/handle/1963/3526401233nas a2200133 4500008004100000245007600041210006900117300001200186490000700198520080200205100002301007700002301030856004601053 2018 eng d00aPositive subharmonic solutions to nonlinear ODEs with indefinite weight0 aPositive subharmonic solutions to nonlinear ODEs with indefinite a17500210 v203 aWe 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).

1 aBoscaggin, Alberto1 aFeltrin, Guglielmo uhttps://doi.org/10.1142/S021919971750021300961nas a2200133 4500008004100000245014200041210006900183260003100252520042800283100002300711700002300734700001900757856005100776 2016 en d00aPairs of positive periodic solutions of nonlinear ODEs with indefinite weight: a topological degree approach for the super-sublinear case0 aPairs of positive periodic solutions of nonlinear ODEs with inde bCambridge University Press3 aWe 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.

1 aBoscaggin, Alberto1 aFeltrin, Guglielmo1 aZanolin, Fabio uhttp://urania.sissa.it/xmlui/handle/1963/3526200555nas a2200133 4500008004100000245009600041210006900137260003700206300001200243490000700255100002300262700001900285856011700304 2013 eng d00aPairs of nodal solutions for a class of nonlinear problems with one-sided growth conditions0 aPairs of nodal solutions for a class of nonlinear problems with bAdvanced Nonlinear Studies, Inc. a13–530 v131 aBoscaggin, Alberto1 aZanolin, Fabio uhttps://www.math.sissa.it/publication/pairs-nodal-solutions-class-nonlinear-problems-one-sided-growth-conditions01133nas a2200157 4500008004100000022001400041245008000055210007300135260000800208300001400216490000700230520064600237100002300883700002300906856004600929 2013 eng d a1420-900400aPlanar Hamiltonian systems at resonance: the Ahmad–Lazer–Paul condition0 aPlanar Hamiltonian systems at resonance the Ahmad–Lazer–Paul con cJun a825–8430 v203 aWe 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.

1 aBoscaggin, Alberto1 aGarrione, Maurizio uhttps://doi.org/10.1007/s00030-012-0181-201092nas a2200205 4500008004100000022001400041245010400055210006900159300000700228490000700235520037600242653003000618653003400648653002300682653003700705653002600742100002300768700001900791856007600810 2013 eng d a1078-094700aSubharmonic solutions for nonlinear second order equations in presence of lower and upper solutions0 aSubharmonic solutions for nonlinear second order equations in pr a890 v333 aWe 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.

10alower and upper solutions10aparameter dependent equations10aPeriodic solutions10aPoincaré-Birkhoff twist theorem10asubharmonic solutions1 aBoscaggin, Alberto1 aZanolin, Fabio uhttp://aimsciences.org//article/id/3638a93e-4f3e-4146-a927-3e8a64e6863f00559nas a2200121 4500008004100000245012000041210006900161260003700230300001400267490000700281100002300288856012600311 2012 eng d00aOne-signed harmonic solutions and sign-changing subharmonic solutions to scalar second order differential equations0 aOnesigned harmonic solutions and signchanging subharmonic soluti bAdvanced Nonlinear Studies, Inc. a445–4630 v121 aBoscaggin, Alberto uhttps://www.math.sissa.it/publication/one-signed-harmonic-solutions-and-sign-changing-subharmonic-solutions-scalar-second01215nas a2200193 4500008004100000022001400041245010000055210006900155300001600224490000800240520056000248653002000808653002500828653003200853653002300885100002300908700001900931856007100950 2012 eng d a0022-039600aPairs of positive periodic solutions of second order nonlinear equations with indefinite weight0 aPairs of positive periodic solutions of second order nonlinear e a2900 - 29210 v2523 aWe 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.

10aCritical points10aNecessary conditions10aPairs of positive solutions10aPeriodic solutions1 aBoscaggin, Alberto1 aZanolin, Fabio uhttp://www.sciencedirect.com/science/article/pii/S002203961100389500752nas a2200133 4500008004100000245006500041210006500106260005100171300001400222490000700236520025200243100002300495856010000518 2012 eng d00aPeriodic solutions to superlinear planar Hamiltonian systems0 aPeriodic solutions to superlinear planar Hamiltonian systems bEuropean Mathematical Society Publishing House a127–1410 v693 aWe 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.

1 aBoscaggin, Alberto uhttps://www.math.sissa.it/publication/periodic-solutions-superlinear-planar-hamiltonian-systems01126nas a2200193 4500008004100000022001400041245013400055210006900189300001600258490000800274520044900282653002100731653001800752653003200770653001700802100002300819700001900842856007100861 2012 eng d a0022-039600aPositive periodic solutions of second order nonlinear equations with indefinite weight: Multiplicity results and complex dynamics0 aPositive periodic solutions of second order nonlinear equations a2922 - 29500 v2523 aWe 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.

10aComplex dynamics10aPoincaré map10aPositive periodic solutions10aSubharmonics1 aBoscaggin, Alberto1 aZanolin, Fabio uhttp://www.sciencedirect.com/science/article/pii/S002203961100388300897nas a2200181 4500008004100000022001400041245008800055210006900143300001400212490000800226520028400234653002200518653003800540653002300578653002000601100002300621856007100644 2011 eng d a0022-247X00aA note on a superlinear indefinite Neumann problem with multiple positive solutions0 anote on a superlinear indefinite Neumann problem with multiple p a259 - 2680 v3773 aWe 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.

10aIndefinite weight10aNonlinear boundary value problems10apositive solutions10aShooting method1 aBoscaggin, Alberto uhttp://www.sciencedirect.com/science/article/pii/S0022247X1000879601464nas a2200193 4500008004100000022001400041245012500055210006900180300001600249490000700265520078200272653003201054653003301086653001401119653002001133100002301153700002301176856007101199 2011 eng d a0362-546X00aResonance and rotation numbers for planar Hamiltonian systems: Multiplicity results via the Poincaré–Birkhoff theorem0 aResonance and rotation numbers for planar Hamiltonian systems Mu a4166 - 41850 v743 aIn 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.

10aMultiple periodic solutions10aPoincaré–Birkhoff theorem10aResonance10aRotation number1 aBoscaggin, Alberto1 aGarrione, Maurizio uhttp://www.sciencedirect.com/science/article/pii/S0362546X1100181700512nas a2200121 4500008004100000245008400041210006900125260003700194300001300231490000700244100002300251856011600274 2011 eng d00aSubharmonic solutions of planar Hamiltonian systems: a rotation number approach0 aSubharmonic solutions of planar Hamiltonian systems a rotation n bAdvanced Nonlinear Studies, Inc. a77–1030 v111 aBoscaggin, Alberto uhttps://www.math.sissa.it/publication/subharmonic-solutions-planar-hamiltonian-systems-rotation-number-approach00785nas a2200121 4500008004100000245009300041210006900134300001400203490000700217520028800224100002300512856012800535 2011 eng d00aSubharmonic solutions of planar Hamiltonian systems via the Poincaré́-Birkhoff theorem0 aSubharmonic solutions of planar Hamiltonian systems via the Poin a115–1220 v663 aWe 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.

1 aBoscaggin, Alberto uhttps://www.math.sissa.it/publication/subharmonic-solutions-planar-hamiltonian-systems-poincar%C3%A9%CC%81-birkhoff-theorem