Positive solutions for some nonlinear elliptic systems in exterior domains of \(\mathbb R^2\) (Q437530)

Summary: 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 \(\Delta u = p(x)u^\alpha v^r, \Delta v = q(x)u^s v^\beta\) in an exterior domain \(D\) of \(\mathbb R^2\), subject to some Dirichlet conditions, where \(\alpha \geq 1, \beta \geq 1, r \geq 0, s \geq 0\) and the potentials \(p, q\) are nonnegative and satisfy some hypotheses related to the Kato class \(K(D)\).