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Abstract

We investigate the existence and qualitative behavior of entire positive radial solutions to the semilinear elliptic system% \begin{equation*} \Delta u=p(|x|)\,g(v),\qquad \Delta v=q(|x|)\,f(u),\qquad x\in \mathbb{R}% ^{n},\ n\geq 3, \end{equation*}% under finite Keller--Osserman-type integral conditions on the nonlinearities $f$ and $g$, and integrability constraints on the radial weights $p$ and $q$% . The nonlinearities are assumed continuous on $[0,\infty )$, differentiable on $(0,\infty )$, vanish at the origin, and are strictly positive elsewhere, with% \begin{equation*} \int_{1}^{\infty }\frac{dt}{g(f(t))}<\infty ,\qquad \int_{1}^{\infty }\frac{% dt}{f(g(t))}<\infty . \end{equation*}% The weights satisfy $\int_{0}^{\infty }s\,p(s)\,ds<\infty $, $% \int_{0}^{\infty }s\,q(s)\,ds<\infty $, and $\min (p,q)$ is not compactly supported. Within this framework, we establish: (i) the existence of infinitely many entire positive radial solutions for a nonempty set of central values; (ii) closedness of the set of admissible central values; and (iii) largeness (blow-up at infinity) of solutions corresponding to boundary points of this set. The approach is based on a novel subharmonic functional tailored to the reciprocal integral conditions, extending classical Keller--Osserman theory to a broad class of coupled systems with general nonlinearities and weight functions.