Opuscula Math. 36, no. 5 (2016), 613-629

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.
  • Received: 2016-03-27.
  • Revised: 2016-04-11.
  • Accepted: 2016-04-15.
  • Published online: 2016-06-29.
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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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