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/1963/3526400564nas a2200133 4500008004100000245012900041210006900170260008500239300001400324490000700338100002300345700001900368856004300387 2017 eng d00aAn application of coincidence degree theory to cyclic feedback type systems associated with nonlinear differential operators0 aapplication of coincidence degree theory to cyclic feedback type bNicolaus Copernicus University, Juliusz P. Schauder Centre for Nonlinear Studies a683–7260 v501 aFeltrin, Guglielmo1 aZanolin, Fabio uhttps://doi.org/10.12775/TMNA.2017.03801598nas a2200217 4500008004100000022001400041245010600055210006900161300001600230490000800246520083500254653002301089653002501112653003601137653003201173653002601205653003601231100002301267700001901290856007101309 2017 eng d a0022-039600aMultiplicity of positive periodic solutions in the superlinear indefinite case via coincidence degree0 aMultiplicity of positive periodic solutions in the superlinear i a4255 - 42910 v2623 aWe study the periodic boundary value problem associated with the second order nonlinear differential equationu″+cu′+(a+(t)−μa−(t))g(u)=0, where g(u) has superlinear growth at zero and at infinity, a(t) is a periodic sign-changing weight, c∈R and μ>0 is a real parameter. Our model includes (for c=0) the so-called nonlinear Hill's equation. We prove the existence of 2m−1 positive solutions when a(t) has m positive humps separated by m negative ones (in a periodicity interval) and μ is sufficiently large, thus giving a complete solution to a problem raised by G.J. Butler in 1976. The proof is based on Mawhin's coincidence degree defined in open (possibly unbounded) sets and applies also to Neumann boundary conditions. Our method also provides a topological approach to detect subharmonic solutions.

10aCoincidence degree10aMultiplicity results10aNeumann boundary value problems10aPositive periodic solutions10asubharmonic solutions10aSuperlinear indefinite problems1 aFeltrin, Guglielmo1 aZanolin, Fabio uhttp://www.sciencedirect.com/science/article/pii/S002203961730021900961nas 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/1963/3526201084nas a2200121 4500008004100000245013700041210006900178260002300247520059900270100002300869700001900892856005100911 2015 en d00aExistence of positive solutions in the superlinear case via coincidence degree: the Neumann and the periodic boundary value problems0 aExistence of positive solutions in the superlinear case via coin bKhayyam Publishing3 aWe prove the existence of positive periodic solutions for the second order nonlinear equation u'' + a(x) g(u) = 0, where g(u) has superlinear growth at zero and at infinity. The weight function a(x) is allowed to change its sign. Necessary and sufficient conditions for the existence of nontrivial solutions are obtained. The proof is based on Mawhin's coincidence degree and applies also to Neumann boundary conditions. Applications are given to the search of positive solutions for a nonlinear PDE in annular domains and for a periodic problem associated to a non-Hamiltonian equation.

1 aFeltrin, Guglielmo1 aZanolin, Fabio uhttp://projecteuclid.org/euclid.ade/143506451801194nas a2200121 4500008004100000245008200041210006900123260001300192520077400205100002300979700001901002856005101021 2015 en d00aMultiple positive solutions for a superlinear problem: a topological approach0 aMultiple positive solutions for a superlinear problem a topologi bElsevier3 aWe study the multiplicity of positive solutions for a two-point boundary value problem associated to the nonlinear second order equation u''+f(x,u)=0. We allow x ↦ f(x,s) to change its sign in order to cover the case of scalar equations with indefinite weight. Roughly speaking, our main assumptions require that f(x,s)/s is below λ_1 as s→0^+ and above λ_1 as s→+∞. In particular, we can deal with the situation in which f(x,s) has a superlinear growth at zero and at infinity. We propose a new approach based on the topological degree which provides the multiplicity of solutions. Applications are given for u'' + a(x) g(u) = 0, where we prove the existence of 2^n-1 positive solutions when a(x) has n positive humps and a^-(x) is sufficiently large.

1 aFeltrin, Guglielmo1 aZanolin, Fabio uhttp://urania.sissa.it/xmlui/handle/1963/3514700555nas 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-conditions01092nas 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-a927-3e8a64e6863f01215nas 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/S002203961100389501126nas 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/S0022039611003883