Positive solution for asymptotically linear elliptic systems

In this paper, we show that semilinear elliptic systems of the form 1\documentclass{article}\footskip=0pc\usepackage{amssymb}\usepackage[mathscr]{euscript}\pagestyle{empty}\begin{document}\begin{eqnarray*}\left\{\begin{array}{ll}-(\Delta u-u)+\mathrm{\mu}(\Delta v-v)=g(x,v), & x \in {\mathbb{R}}^N\\-(\Delta v-v)+\lambda(\Delta u-u)=f(x,u), & x \in {\mathbb{R}}^N\end{array}\right.\end{eqnarray*}\end{document} possess at least one positive solution pair ( u, v )∈ H 1 (ℝ N )× H 1 (ℝ N ), where λ and µ are nonnegative numbers, f ( x, t ) and g ( x, t ) are continuous functions on ℝ N ×ℝ and asymptotically linear as t →+∞. Copyright © 2009 John Wiley & Sons, Ltd.