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

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• 作者：Guglielmo Feltrin^a ; ¹ ; ^{guglielmo.feltrin@sissa.it} ; ^{guglielmo.feltrin@umons.ac.be} ; Fabio Zanolin^b ; ^{fabio.zanolin@uniud.it}
• 关键词：34B18 ; 34B15 ; 34C25 ; 47H11
• 刊名：Journal of Differential Equations
• 出版年：2017
• 出版时间：15 April 2017
• 年：2017
• 卷：262
• 期：8
• 页码：4255-4291
• 全文大小：760 K
• 卷排序：262

文摘

We study the periodic boundary value problem associated with the second order nonlinear differential equationu″+cu′+(a+(t)−μa−(t))g(u)=0,u″+cu′+(a+(t)−μa−(t))g(u)=0, where g(u)g(u) has superlinear growth at zero and at infinity, a(t)a(t) is a periodic sign-changing weight, c∈Rc∈R and μ>0μ>0 is a real parameter. Our model includes (for c=0c=0) the so-called nonlinear Hill's equation. We prove the existence of 2m−12m−1 positive solutions when a(t)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.