Well-Posedness and Monotone Analysis for a Coupled Sublinear Lane--Emden--Fowler System on Bounded Domains

We investigate a coupled system of elliptic equations of Lane--Emden--Fowler type on a bounded domain $\Omega \subset \mathbb{R}^n$ ($n \geq 1$) with homogeneous Dirichlet boundary conditions. The system is characterized by sublinear power-law reaction terms $0 < \alpha, \beta < 1$ and includes a fidelity regularization component. Due to the non-gradient structure of the coupling, we employ the method of sub- and supersolutions and a monotone iteration scheme to establish the existence of positive solutions. We prove that the system admits a unique positive solution $(u,v) \in C^{1,\gamma}($ for some $\gamma \in (0,1)$, and we demonstrate the continuous dependence of the solution on the data. For the discrete case, we establish the monotone convergence of a fixed-point algorithm by verifying the conditions of Krasnosel'ski\u{\i}'s theorem for monotone sub-homogeneous operators. This work provides a rigorous mathematical foundation for coupled reaction-diffusion models where traditional variational minimization is not directly applicable.