Opuscula Math. 35, no. 1 (2015), 21-36
http://dx.doi.org/10.7494/OpMath.2015.35.1.21

 
Opuscula Mathematica

Asymptotic behavior of positive solutions of a semilinear Dirichlet problem in the annulus

Safa Dridi
Bilel Khamessi

Abstract. In this paper, we establish existence and asymptotic behavior of a positive classical solution to the following semilinear boundary value problem: \[-\Delta u=q(x)u^{\sigma }\;\text{in}\;\Omega,\quad u_{|\partial\Omega}=0.\] Here \(\Omega\) is an annulus in \(\mathbb{R}^{n}\), \(n\geq 3\), \(\sigma \lt 1\) and \(q\) is a positive function in \(\mathcal{C}_{loc}^{\gamma }(\Omega )\), \(0\lt\gamma \lt 1\), satisfying some appropriate assumptions related to Karamata regular variation theory. Our arguments combine a method of sub- and supersolutions with Karamata regular variation theory.

Keywords: asymptotic behavior, Dirichlet problem, Karamata function, subsolution, supersolution.

Mathematics Subject Classification: 31C15, 34B27, 35K10.

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  • Safa Dridi
  • Campus Universitaire, Faculté des Sciences de Tunis, Département de Mathématiques, 2092 Tunis, Tunisia
  • Bilel Khamessi
  • Campus Universitaire, Faculté des Sciences de Tunis, Département de Mathématiques, 2092 Tunis, Tunisia
  • Communicated by Vicentiu D. Radulescu.
  • Received: 2014-02-18.
  • Revised: 2014-05-05.
  • Accepted: 2014-05-05.
  • Published online: 2014-11-12.
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Cite this article as:
Safa Dridi, Bilel Khamessi, Asymptotic behavior of positive solutions of a semilinear Dirichlet problem in the annulus, Opuscula Math. 35, no. 1 (2015), 21-36, http://dx.doi.org/10.7494/OpMath.2015.35.1.21

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