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

We study positive radial solutions of the weighted cooperative system $\Delta u=p(|x|)f(u,v)$, $\Delta v=q(|x|)g(u,v)$ in $\mathbb{R}^{n}$, $n\geq3$, when the Newtonian potentials of the nonnegative weights are finite and the nonlinear source possesses a Keller--Osserman lower envelope.

A threshold set of central data is introduced and its order and compactness properties are used to construct an entire solution on its interior boundary.

A decreasing nonlinear transform applied to $W=u+v$ yields an explicit lower estimate in terms of the tail Newtonian potential and proves that $W(r)\to\infty$.

Consequently at least one component is large.

We also identify the additional hypothesis genuinely needed for componentwise blow-up: diagonal Keller--Osserman integrability alone is insufficient, whereas a two-sided balance of the weighted reaction channels forces $u(r),v(r)\to\infty$.

Two independent proofs of this transfer principle are given, one based on radial fluxes and one on Newtonian potentials.

Finally, the PDE system is derived from an isothermal two-species diffusion--reaction balance in a heterogeneous porous catalyst, including nondimensional parameters and the interpretation of the whole-space idealization.