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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