Large solutions for cooperative logistic systems: existence and uniqueness in star-shaped domains

We extend Theorem 1.1 of [J. Math. Anal. Appl. 435 (2016), 1738-1752] to show the uniqueness of large solutions for the system of (1) in star-shaped domains. This result is due to the maximum principle for cooperative systems of [J. López-Gómez and M. Molina-Meyer, Diff. Int. Eq. 7 (1994), 383-398], which allows us to establish the uniqueness without invoking to the blow-up rates of the solutions.