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Existence and \(L_\infty\) estimates for a class of singular ordinary differential equations. (English) Zbl 1083.34016
The author considers Dirichet’s boundary value problem on \(I:=[0,1]\) for \[ (\Phi u')' + \Phi \times f \circ u = 0. \] Assuming \(\Phi(t) \geq m > 0\) and several other assumptions concerning \(\Phi\) and \(f\), the author postulates that there should exist a nonnegative \(v \in H_0^1(I)\) such that \(\frac{1}{2} \int_{0}^{1} \Phi {v'}^2 \,dt < \int_{0}^{1} \Phi F \circ v \,dt\) with \(F(t) := \int_{0}^{t} f(s) \,ds \). He is then able to prove the existence of a positive solution as well as the \(L^\infty\)-estimate \(\max u \leq \| v\| _\infty\). The result is extended by an approximation procedure to corresponding boundary value problems for which \(\Phi\) may cause a singularity at \(t= 0\).

MSC:
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B16 Singular nonlinear boundary value problems for ordinary differential equations
34C11 Growth and boundedness of solutions to ordinary differential equations
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References:
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