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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• Syrine Masmoudi
• Département de Mathématiques, Faculté des Sciences de Tunis, Campus Universitaire, 2092 Tunis, Tunisia
• Samia Zermani
• Département de Mathématiques, Faculté des Sciences de Tunis, Campus Universitaire, 2092 Tunis, Tunisia
• Communicated by Vicentiu D. Radulescu.
• Revised: 2016-04-11.
• Accepted: 2016-04-15.
• Published online: 2016-06-29.