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Existence of positive solutions of a superlinear boundary value problem with indefinite weight
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 $g\colon\mathopen{[}0,+\infty\mathclose{[}\to\mathbb{R}$ 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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