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.

Full text (pdf)

  1. R.P. Agarwal, D. O'Regan, Nonlinear superlinear singular and nonsingular second order boundary value problems, J. Differential Equations 143 (1998), 60-95.
  2. R. Alsaedi, H. Mâagli, V. Radulescu, N. Zeddini, Asymptotic behaviour of positive large solutions of quasilinear logistic problems, Electron. J. Qual. Theory Differ. Equ. 28 (2015), 1-15.
  3. I. Bachar, H. Mâagli, Existence and global asymptotic behavior of positive solutions for combined second-order differential equations on the half-line, Adv. Nonlinear Anal. (DOI: 10.1515/anona-2015-0078). http://dx.doi.org/10.1515/anona-2015-0078.
  4. L.E. Bobisud, D. O'Regan,Positive solutions for a class of nonlinear singular boundary value problems at resonance, J. Math. Anal. Appl. 184 (1994), 263-284.
  5. A. Callegari, A. Nachman, A nonlinear singular boundary value problem in the theory of pseudoplastic fluids, SIAM J. Appl. Math. 38 (1980), 275-281.
  6. R. Chemmam, H. Mâagli, S. Masmoudi, M. Zribi, Combined effects in nonlinear singular elliptic problems in a bounded domain, Adv. Nonlinear Anal. 51 (2012), 301-318.
  7. X. Cheng, G. Dai, Positive solutions of sub-superlinear Sturm-Liouville problems, Appl. Math. Comput. 261 (2015), 351-359.
  8. F. Cîrstea, V. Radulescu, Uniqueness of the blow-up boundary solution of logistic equations with absorption, C. R. Acad. Sci. Paris, Ser. I 335 (2002), 447-452.
  9. F. Cîrstea, V. Radulescu, Boundary blow-up in nonlinear elliptic equations of Bieberbach-Rademacher type, Trans. Amer. Math. Soc. 359 (2007), 3275-3286.
  10. R. Dalmasso, On singular nonlinear elliptic problems of second and fourth orders, Bull. Sci. Math. 116 (1992), 95-110.
  11. S. Dridi, B. Khamessi, S. Turki, Z. Zine El Abidine, Asymptotic behavior of positive solutions of a semilinear Dirichlet problem, Nonlinear Stud. 22 (2015) 1, 1-17.
  12. R. Kannan, D. O'Regan, A note on singular boundary value problems with solutions in weighted spaces, Nonlinear Anal. 37 (1999), 791-796.
  13. H. Li, J. Sun, Positive solutions of sublinear Sturm-Liouville problems with changing sign nonlinearity, Comput. Math. Appl. 58 (2009), 1808-1815.
  14. C.D. Luning, W.L. Perry, Positive solutions of negative exponent generalized Emden Fowler boundary value problems, SIAM J. Math. Anal. 12 (1981), 874-879.
  15. H. Mâagli, Asymptotic behavior of positive solutions of a semilinear Dirichlet problem, Nonlinear Anal. 74 (2011), 2941-2947.
  16. H. Mâagli, S. Masmoudi, Sur les solutions d'un opérateur différentiel singulier semi-linéaire, Potential Anal. 10 (1999), 289-304.
  17. H. Mâagli, S. Ben Othman, S. Dridi, Existence and asymptotic behavior of ground state solutions of semilinear elliptic system, Adv. Nonlinear Anal. (DOI: 10.1515/anona-2015-0157). http://dx.doi.org/10.1515/anona-2015-0157.
  18. V. Radulescu, Singular phenomena in nonlinear elliptic problems. From blow-up boundary solutions to equations with singular nonlinearities, [in:] Handbook of Differential Equations: Stationary Partial Differential Equations, Vol. 4 (Michel Chipot, Editor), North-Holland Elsevier Science, Amsterdam, 2007, 483-591.
  19. E. Seneta, Regularly Varying Functions, Lectures Notes in Mathematics 508, Springer-Verlag, Berlin-New York, 1976.
  20. Y. Sun, L. Liu, Y. Je Cho, Positive solutions of singular nonlinear Sturm-Liouville boundary value problems, ANZIAM J. 45 (2004), 557-571.
  21. S.D. Taliaferro, A nonlinear singular boundary value problem, Nonlinear Anal. 3 (1979) 6, 897-904.
  22. H. Usami, On a singular elliptic boundary value problem in a ball, Nonlinear Anal. 13 (1989), 1163-1170.
  23. N. Yazidi, Monotone method for singular Neumann problem, Nonlinear Anal. 49 (2002), 589-602.
  24. Y. Zhang, Positive solutions of singular sublinear Emden-Fowler boundary value problems, J. Math. Anal. Appl. 185 (1994), 215-222.
  • 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.
Opuscula Mathematica - cover

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

Download this article's citation as:
a .bib file (BibTeX),
a .ris file (RefMan),
a .enw file (EndNote)
or export to RefWorks.

In accordance with EU legislation we advise you this website uses cookies to allow us to see how the site is used. All data is anonymized.
All recent versions of popular browsers give users a level of control over cookies. Users can set their browsers to accept or reject all, or certain, cookies.