Multiplicity of positive periodic solutions in the superlinear indefinite case via coincidence degree

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:262

Article

Feltrin, Guglielmo^1,3  Zanolin, Fabio² 

[1] SISSA Int Sch Adv Studies, Via Bonomea 265, I-34136 Trieste, Italy

[2] Univ Udine, Dept Math Comp Sci & Phys, Via Sci 206, I-33100 Udine, Italy

[3] Univ Mons, Dept Math, Pl Parc 20, B-7000 Mons, Belgium

关键词: Superlinear indefinite problems;    Positive periodic solutions;    Multiplicity results;    Subharmonic solutions;    Neumann boundary value problems;    Coincidence degree;   

DOI  :  10.1016/j.jde.2017.01.009

来源: Elsevier

【 摘 要 】

We study the periodic boundary value problem associated with the second order nonlinear differential equation u '' +cu ' + (a(+)(t) - mu a(-)(t))g(u) = 0, where g (u) has superlinear growth at zero and at infinity, a(t) is a periodic sign-changing weight, c is an element of R and mu > 0 is a real parameter. Our model includes (for c = 0) the so-called nonlinear Hill's equation. We prove the existence of 2(m) - 1 positive solutions when a(t) has m positive humps separated by m negative ones (in a periodicity interval) and mu 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. (C)2017 Elsevier Inc. All rights reserved.

【 授权许可】

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