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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)

34B15 Nonlinear boundary value problems for ordinary differential equations
34B24 Sturm-Liouville theory
Full Text: DOI
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