TY - JOUR T1 - Multiplicity of positive periodic solutions in the superlinear indefinite case via coincidence degree JF - Journal of Differential Equations Y1 - 2017 A1 - Guglielmo Feltrin A1 - Fabio Zanolin KW - Coincidence degree KW - Multiplicity results KW - Neumann boundary value problems KW - Positive periodic solutions KW - subharmonic solutions KW - Superlinear indefinite problems AB -

We 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.

VL - 262 UR - http://www.sciencedirect.com/science/article/pii/S0022039617300219 ER -