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.

Positive solutions for super-sublinear indefinite problems: high multiplicity results via coincidence degree

Alberto Boscaggin;Guglielmo Feltrin;Fabio Zanolin
2018-01-01

Abstract

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.
2018
370
2
791
845
https://doi.org/10.1090/tran/6992
https://arxiv.org/abs/1512.07138
boundary value problems, positive solutions, indefinite weight, super-sublinear nonlinearity, multiplicity results, symbolic dynamics, coincidence degree
Alberto, Boscaggin; Guglielmo, Feltrin; Fabio, Zanolin
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2318/1542245
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