Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
5774250 | Journal of Differential Equations | 2017 | 37 Pages |
Abstract
We study the periodic boundary value problem associated with the second order nonlinear differential equationuâ³+cuâ²+(a+(t)âμaâ(t))g(u)=0, where g(u) has superlinear growth at zero and at infinity, a(t) is a periodic sign-changing weight, câR 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 2mâ1 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.
Keywords
Related Topics
Physical Sciences and Engineering
Mathematics
Analysis
Authors
Guglielmo Feltrin, Fabio Zanolin,