We deal with the existence of positive solutions for a two-point boundary value problem associated with the nonlinear second order equation u''+a(x)g(u)=0. The weight a(x) is allowed to change sign. We assume that the function g:[0,+∞[→ℝ is continuous, g(0)=0 and satisfies suitable growth conditions, including the superlinear case g(s)=s^p, with p>1. In particular we suppose that g(s)/s is large near infinity, but we do not require that g(s) is non-negative in a neighborhood of zero. Using a topological approach based on the Leray-Schauder degree we obtain a result of existence of at least a positive solution that improves previous existence theorems.
Existence of positive solutions of a superlinear boundary value problem with indefinite weight
Feltrin, Guglielmo
2015-01-01
Abstract
We deal with the existence of positive solutions for a two-point boundary value problem associated with the nonlinear second order equation u''+a(x)g(u)=0. The weight a(x) is allowed to change sign. We assume that the function g:[0,+∞[→ℝ is continuous, g(0)=0 and satisfies suitable growth conditions, including the superlinear case g(s)=s^p, with p>1. In particular we suppose that g(s)/s is large near infinity, but we do not require that g(s) is non-negative in a neighborhood of zero. Using a topological approach based on the Leray-Schauder degree we obtain a result of existence of at least a positive solution that improves previous existence theorems.
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