Opuscula Mathematica
Opuscula Math. 36, no. 3 (2016), 315-336

Existence and asymptotic behavior of positive solutions of a semilinear elliptic system in a bounded domain

Majda Chaieb
Abdelwaheb Dhifli
Samia Zermani
Abstract. Let \(\Omega\) be a bounded domain in \(\mathbb{R}^{n}\) (\(n\geq 2\)) with a smooth boundary \(\partial \Omega\). We discuss in this paper the existence and the asymptotic behavior of positive solutions of the following semilinear elliptic system \[\begin{aligned} -\Delta u&=a_{1}(x)u^{\alpha}v^{r}\quad\text{in}\;\Omega ,\;\;\,u|_{\partial\Omega}=0,\\ -\Delta v&=a_{2}(x)v^{\beta}u^{s}\quad\text{in}\;\Omega ,\;\;\,v|_{\partial\Omega }=0.\end{aligned}\] Here \(r,s\in \mathbb{R}\), \(\alpha,\beta \lt 1\) such that \(\gamma :=(1-\alpha)(1-\beta)-rs\gt 0\) and the functions \(a_{i}\) (\(i=1,2\)) are nonnegative and satisfy some appropriate conditions with reference to Karamata regular variation theory.
Keywords: semilinear elliptic system, asymptotic behavior, Karamata class, sub-super solution.
Mathematics Subject Classification: 31B25, 34B15, 34B18, 34B27.
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About the authors ▼
Majda Chaieb
Faculté des Sciences de Tunis, Département de Mathématiques, Campus Universitaire, 2092 Tunis, Tunisia
Abdelwaheb Dhifli
Faculté des Sciences de Tunis, Département de Mathématiques, Campus Universitaire, 2092 Tunis, Tunisia
Samia Zermani
Faculté des Sciences de Tunis, Département de Mathématiques, Campus Universitaire, 2092 Tunis, Tunisia
About the article ▼
Communicated by Vicentiu D. Radulescu.
Received: 2014-06-10.
Revised: 2015-10-27.
Accepted: 2015-11-15.
Published online: 2016-02-21.
Cite this article as ▼
Majda Chaieb, Abdelwaheb Dhifli, Samia Zermani, Existence and asymptotic behavior of positive solutions of a semilinear elliptic system in a bounded domain, Opuscula Math. 36, no. 3 (2016), 315-336, http://dx.doi.org/10.7494/OpMath.2016.36.3.315.
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