two solutions must be sought for a supersolution and the other for subsolution. We can reduce ourselves to seek a positive solution for the problem supersolution U> 0 ie P (D) U> F (x, U) on Omega U> 0 on the edge of Omega for subsolution V

What is the problem? I assume it is a boundary value problem. What do you know about the forward differential equation? This will be crucial for trying to apply a topological shooting argument.

Thank you for your reply. Actually, I am unable to prove the existence results of the solution of coupled non-linear differential equations arising in non-Newtonian Reiner-Rivlin flow above a rough rotating disk. The constitutive equations are highly non-linear and coupled pair. I am attaching my problem in Pdf format.