New Insights on Keller–Osserman Conditions for Semilinear Systems

In this article, we consider a semilinear elliptic system involving gradient terms of the form Δyx+λ1∇yx=pxfyx,zxifx∈Ω,Δzx+λ2∇zx=qxgyxifx∈Ω, where λ1, λ2∈0,∞, Ω is either a ball of radius R>0 or the entire space RN. Based on certain standard assumptions regarding the potential functions p and q, we introduce new conditions on the nonlinearities f and g to investigate the existence of entire large solutions for the given system. The method employed is successive approximation. Additionally, for specific cases of p, q, f and g, we employ Python code to plot the graph of both the numerical solution and the exact solution.