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A methodology is presented for the numerical solution of nonlinear elliptic systems in unbounded domains, consisting of three elements. First, the problem is posed on a finite domain by means of a proper nonlinear change of variables. The compressed domain is then discretised, regardless of its final shape, via the radial basis function partition of unity method. Finally, the system of nonlinear algebraic collocation equations is solved with the trust-region algorithm, taking advantage of analytically derived Jacobians. We validate the methodology on a benchmark of computational fluid mechanics: the steady viscous flow past a circular cylinder. The resulting flow characteristics compare very well with the literature. Then, we stress-test the methodology on less smooth obstacles—rounded and sharp square cylinders. As expected, in the latter scenario the solution is polluted by spurious oscillations, owing to the presence of boundary singularities. • Infinite domain compression removes the need for artificial computational domain and the hard-to-quantify associated error. • Nonlinear elliptic PDEs, particularly systems thereof, are very rare in the RBF literature. • RBF-PU allows for handling systems of nonlinear PDEs in strong form on irregular domains without loss of performance. • RBF-PU's accurate derivates lead to a compact, sparse, easy-to-derive analytical Jacobian. • Detailed transition from circular to square cylinder investigated numerically, showing good agreement with published results. [ABSTRACT FROM AUTHOR]