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Singular Dirichlet boundary value problems. I: Superlinear and nonresonant case. (English) Zbl 0884.34028
This paper establishes existence results for the Dirichlet second order boundary value problem $y''+ \mu a(t)y = f(t,y)\quad\text{a.e. on }[0,1]$
$y(0) = y(1) = 0$ where $$\mu$$ is such that $y''+ \mu a(t)y = 0\quad\text{a.e. on }[0,1]$
$y(0) = y(1) = 0$ has only the trivial solution. $$f:[0,1]\times \mathbb{R}\rightarrow \mathbb{R}$$ is a Carathéodory function, $$a\in L^1_{loc}(0,1)$$ with $$a>0$$ a. e. on $$[0,1]$$ and $$\int_0^1x(1-x)a(x)dx<\infty.$$ A first result concerns the case when $$f$$ is superlinear and a second one is a nonresonant type result. In preparation an appropriate eigenvalue problem is considered.
Reviewer: M.Goebel (Halle)

##### MSC:
 34B15 Nonlinear boundary value problems for ordinary differential equations 34B24 Sturm-Liouville theory
##### Keywords:
existence; singular; nonresonance; superlinear
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##### References:
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