We deal with positive solutions for the Neumann boundary value problem associated with the scalar second order ODE u″ + q(t)g(u) = 0, t ϵ [0, T]; where g : [0,+∞[→ R is positive on ]0,+ ∞ [ and q(t) is an indefinite weight. Complementary to previous investigations in the case ∫T0 q(t) < 0, we provide existence results for a suitable class of weights having (small) positive mean, when g′(u) < 0 at infinity. Our proof relies on a shooting argument for a suitable equivalent planar system of the type x′ = y, y′ = h(x)y2 + q(t); with h(x) a continuous function defined on the whole real line.
Boscaggin, A., Garrione, M. (2016). Positive solutions to indefinite neumann problems when the weight has positive average. DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS, 36(10), 5231-5244 [10.3934/dcds.2016028].
Positive solutions to indefinite neumann problems when the weight has positive average
GARRIONE, MAURIZIO
2016
Abstract
We deal with positive solutions for the Neumann boundary value problem associated with the scalar second order ODE u″ + q(t)g(u) = 0, t ϵ [0, T]; where g : [0,+∞[→ R is positive on ]0,+ ∞ [ and q(t) is an indefinite weight. Complementary to previous investigations in the case ∫T0 q(t) < 0, we provide existence results for a suitable class of weights having (small) positive mean, when g′(u) < 0 at infinity. Our proof relies on a shooting argument for a suitable equivalent planar system of the type x′ = y, y′ = h(x)y2 + q(t); with h(x) a continuous function defined on the whole real line.
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