Multiple Solutions to Neumann Problems with Indefinite Weight and Bounded Nonlinearities

We study the Neumann boundary value problem for the second order ODE 1 u ″ + ( a + ( t ) - μ a - ( t ) ) g ( u ) = 0 , t ∈ [ 0 , T ] , where g ∈ C 1 ( R ) is a bounded function of constant sign, a + , a - : [ 0 , T ] → R + are the positive/negative part of a sign-changing weight a ( t ) and μ > 0...

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Bibliographic Details
Published in:Journal of dynamics and differential equations 2016-03, Vol.28 (1), p.167-187
Main Authors: Boscaggin, Alberto, Garrione, Maurizio
Format: Article
Language:English
Subjects:
Applications of Mathematics
Mathematics
Mathematics and Statistics
Ordinary Differential Equations
Partial Differential Equations
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Summary:We study the Neumann boundary value problem for the second order ODE 1 u ″ + ( a + ( t ) - μ a - ( t ) ) g ( u ) = 0 , t ∈ [ 0 , T ] , where g ∈ C 1 ( R ) is a bounded function of constant sign, a + , a - : [ 0 , T ] → R + are the positive/negative part of a sign-changing weight a ( t ) and μ > 0 is a real parameter. Depending on the sign of g ′ ( u ) at infinity, we find existence/multiplicity of solutions for μ in a “small” interval near the value μ c = ∫ 0 T a + ( t ) d t ∫ 0 T a - ( t ) d t . The proof exploits a change of variables, transforming the sign-indefinite Eq. ( 1 ) into a forced perturbation of an autonomous planar system, and a shooting argument. Nonexistence results for μ → 0 + and μ → + ∞ are given, as well.
ISSN:1040-7294
1572-9222
DOI:10.1007/s10884-015-9430-5