Opuscula Math. 36, no. 5 (2016), 613-629
http://dx.doi.org/10.7494/OpMath.2016.36.5.613

 
Opuscula Mathematica

Existence and boundary behavior of positive solutions for a Sturm-Liouville problem

Syrine Masmoudi
Samia Zermani

Abstract. In this paper, we discuss existence, uniqueness and boundary behavior of a positive solution to the following nonlinear Sturm-Liouville problem \[\begin{aligned}&\frac{1}{A}(Au^{\prime })^{\prime }+a(t)u^{\sigma}=0\;\;\text{in}\;(0,1),\\ &\lim\limits_{t\to 0}Au^{\prime}(t)=0,\quad u(1)=0,\end{aligned}\] where \(\sigma \lt 1\), \(A\) is a positive differentiable function on \((0,1)\) and \(a\) is a positive measurable function in \((0,1)\) satisfying some appropriate assumptions related to the Karamata class. Our main result is obtained by means of fixed point methods combined with Karamata regular variation theory.

Keywords: nonlinear Sturm-Liouville problem, Green's function, positive solutions, Karamata regular variation theory.

Mathematics Subject Classification: 34B18, 34B27.

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Cite this article as:
Syrine Masmoudi, Samia Zermani, Existence and boundary behavior of positive solutions for a Sturm-Liouville problem, Opuscula Math. 36, no. 5 (2016), 613-629, http://dx.doi.org/10.7494/OpMath.2016.36.5.613

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