Entire Large Solutions for Competitive Semilinear Elliptic Systems with General Nonlinearities Satisfying Keller--Osserman Conditions

We generalize a theorem of Lair concerning the existence of positive entire large solutions to competitive semilinear elliptic systems. While Lair's original result \cite{Lair2025} was established for power-type nonlinearities, our work extends the theory to a broad class of general nonlinearities satisfying a Keller--Osserman-type growth condition. The proof follows the same conceptual framework monotone iteration to construct global positive solutions, reduction to a scalar inequality for the sum of the components, application of a Keller--Osserman transform, and a two-step radial integration argument but replaces the explicit power-law growth with a general monotone envelope function. This approach yields a unified and verifiable criterion for the existence of large solutions in terms of the Keller--Osserman integral, thereby encompassing both critical and supercritical growth regimes within a single analytical setting.