Opuscula Math. 32, no. 4 (2012), 783-795
http://dx.doi.org/10.7494/OpMath.2012.32.4.783

 
Opuscula Mathematica

Existence and asymptotic behavior of positive continuous solutions for a nonlinear elliptic system in the half space

Sameh Turki

Abstract. This paper deals with the existence and the asymptotic behavior of positive continuous solutions of the nonlinear elliptic system \(\Delta u=p(x)u^{\alpha}v^r\), \(\Delta v = q(x)u^s v^{\beta}\), in the half space \(\mathbb{R}^n_+ :=\{x=(x_1,..., x_n)\in \mathbb{R}^n : x_n \gt 0\}\), \(n \geq 2\), where \(\alpha, \beta \gt 1\) and \(r, s \geq 0\). The functions \(p\) and \(q\) are required to satisfy some appropriate conditions related to the Kato class \(K^{\infty}(\mathbb{R}^n_+)\). Our approach is based on potential theory tools and the use of Schauder's fixed point theorem.

Keywords: asymptotic behavior, elliptic system, regular equation.

Mathematics Subject Classification: 34B27, 34J65.

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  • Sameh Turki
  • Faculté des Sciences de Tunis, Département de Mathématiques, Campus Universitaire, 2092 Tunis, Tunisia
  • Received: 2011-11-04.
  • Revised: 2012-02-25.
  • Accepted: 2012-02-27.
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Cite this article as:
Sameh Turki, Existence and asymptotic behavior of positive continuous solutions for a nonlinear elliptic system in the half space, Opuscula Math. 32, no. 4 (2012), 783-795, http://dx.doi.org/10.7494/OpMath.2012.32.4.783

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