Opuscula Math. 36, no. 3 (2016), 315-336
http://dx.doi.org/10.7494/OpMath.2016.36.3.315

Opuscula Mathematica

# 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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• 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
• Communicated by Vicentiu D. Radulescu.
• Revised: 2015-10-27.
• Accepted: 2015-11-15.
• Published online: 2016-02-21.