I am attempting to show existence for a system of $3$ equations (one of which is a quasilinear second-order elliptic PDE), where the form of the PDE depends on the metric since the equation comes from geometry.
If I make an assumption on the exact form of the metric (namely, I assume it to be a small perturbation on another well-known base metric), and then plug this into the PDE to simplify the terms involving gradient and Hessian tensor, no gradient estimates are necessary, so is it possible to show that the system of equations has existence of solutions directly using the implicit function theorem for Banach spaces, or does one need to incorporate the IFT for Banach spaces into a method of continuity argument?
I have a set of equation which is simplified, but there is a function in the equations corresponding to the perturbation.