Liouville‐type results for non‐cooperative elliptic systems in a half‐space

For a class of semilinear elliptic systems, we show that, under quite general assumptions, a Liouville‐type result for the entire space problem extends to the corresponding half‐space Dirichlet problem. This implies a simple criterion for a priori bounds for more general boundary value problems which are also discussed. The class of system we study includes weakly coupled Schrödinger systems, which have attracted great interest in recent years due to their appearance in nonlinear optics and in models for multi‐component mixtures of Bose–Einstein condensates. In this case, using the non‐existence results recently obtained in Tavares, Terracini, Verzini and Weth for non‐trivial entire solutions, we identify new parameter ranges for which a priori bounds hold. The key ideas used in this paper are quite different from earlier methods since we also deal with non‐cooperative systems. In particular, the moving plane method does not apply here.