<p>This work examines the sublinear Lane-Emden-Fowler system in two dimensional exterior domains and subsequently applies the resulting analysis to the case of a rectangular domain. Motivated by earlier scalar results, we extend the analysis to the coupled system case, which, to the best of our knowledge, has not been previously addressed in the literature. Using the Liouville transformation, Schauder’s fixed point theorem, and the sub–supersolution method, we establish the existence of bounded positive solutions under natural integrability conditions on the nonlinear coefficients. Both radial and non-radial settings are treated, in the strictly sublinear regime.</p>