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 Topol. Methods Nonlinear Anal. %D 2017 %T An application of coincidence degree theory to cyclic feedback type systems associated with nonlinear differential operators %A Guglielmo Feltrin %A Fabio Zanolin %B Topol. Methods Nonlinear Anal. %I Nicolaus Copernicus University, Juliusz P. Schauder Centre for Nonlinear Studies %V 50 %P 683–726 %G eng %U https://doi.org/10.12775/TMNA.2017.038 %R 10.12775/TMNA.2017.038 %0 Journal Article %J Journal of Differential Equations %D 2017 %T Multiplicity of positive periodic solutions in the superlinear indefinite case via coincidence degree %A Guglielmo Feltrin %A Fabio Zanolin %K Coincidence degree %K Multiplicity results %K Neumann boundary value problems %K Positive periodic solutions %K subharmonic solutions %K Superlinear indefinite problems %XWe 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.

%B Journal of Differential Equations %V 262 %P 4255 - 4291 %G eng %U http://www.sciencedirect.com/science/article/pii/S0022039617300219 %R https://doi.org/10.1016/j.jde.2017.01.009 %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 Adv. Differential Equations 20 (2015), 937–982. %D 2015 %T Existence of positive solutions in the superlinear case via coincidence degree: the Neumann and the periodic boundary value problems %A Guglielmo Feltrin %A Fabio Zanolin %XWe 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.

%B Adv. Differential Equations 20 (2015), 937–982. %I Khayyam Publishing %G en %U http://projecteuclid.org/euclid.ade/1435064518 %1 35388 %2 Mathematics %4 1 %# MAT/05 %$ Submitted by gfeltrin@sissa.it (gfeltrin@sissa.it) on 2015-12-18T08:49:04Z No. of bitstreams: 1 FeltrinZanolin_ade2015.pdf: 321052 bytes, checksum: 38a45a1515a02fc86751b5f088bbfbfb (MD5) %0 Journal Article %J J. Differential Equations 259 (2015), 925–963. %D 2015 %T Multiple positive solutions for a superlinear problem: a topological approach %A Guglielmo Feltrin %A Fabio Zanolin %XWe 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.

%B J. Differential Equations 259 (2015), 925–963. %I Elsevier %G en %U http://urania.sissa.it/xmlui/handle/1963/35147 %1 35387 %2 Mathematics %4 1 %# MAT/05 %$ Submitted by gfeltrin@sissa.it (gfeltrin@sissa.it) on 2015-12-17T09:31:06Z No. of bitstreams: 1 FeltrinZanolin_jde2015.pdf: 350880 bytes, checksum: 0e329b01081df570863ea2492ffefe0a (MD5) %R 10.1016/j.jde.2015.02.032 %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 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 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 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