Positive Solutions for Some Nonlinear Elliptic Systems in Exterior Domains of ℝ 2

Abstract

Using some potential theory tools and the Schauder fixed point theorem, we prove the existence of positive continuous solutions with a precise global behavior for the competitive semilinear elliptic system $\Delta u=p\left(x\right)u^{\alpha }{v}^r$, $\Delta v=q\left(x\right)u^s{v}^{\beta }$ in an exterior domain $D$ of ${\mathbb{R}}^2$, subject to some Dirichlet conditions, where $\alpha \ge 1$, $\beta \ge 1$, $r\ge 0$, $s\ge 0$ and the potentials $p,q$ are nonnegative and satisfy some hypotheses related to the Kato class $K\left(D\right)$.