Linearization by freezing the coefficients of the main part of the PDE 

Let $\Omega\subset C^0$ a bounded domian in $\mathbb{R}^2$. Let $u\in C^2(\Omega)\cap C(\overline{\Omega})$ be a non negative classical solution of
    $$
(1+x^2)u_{xx}-2xu_{xy}+(1+u)u_{yy}-(1+u^2)u_x+(1+u_x)u_y-u=1\text{ in }\Omega,\\u(x,y)=\frac{\sin^2(x)}{1+y^2}\text{ on }\partial\Omega.
$$
    Show that $(0\leq) y\leq 1$ for $(x,y)\in\Omega$.


I already asked this here: Do I have to use a maximum principle?
But this time my question concerns a special method for solving this, I think it is called something like linearization by freezing the coefficients.
I do not know more about that, but I heard that it works by inserting the assumed solution $u$ in the coefficients of the main part of the PDE and the aim is to get a semi-linear (or linear?) PDE on which one can apply maximum principle.
Do you know something about that method?
 A: The idea behind the "freezing" technique is to introduce a new "source term" $v$ and replace enough appearances of $u$ by $v$ such that the resulting equation is linear - for example in this case we could try 
$$F_v(u)=(1+x^2)u_{xx} - 2xu_{xy}+(1+v)u_{yy} - (1+v^2)u_x + (1-v_x) u_y - u - 1= 0.$$(For certain purposes it may be more fruitful to do something else with the first-order term - it's not immediately obvious what the "correct" choice is to me. For the purpose of establishing the maximum principle it doesn't matter.)
A function $u$ is then a solution of the original PDE if and only if $F_u(u) = 0$; i.e. iff it is a solution of the linear PDE $F_u (\cdot)= 0$. In some cases this could give a way to find solutions: if the linear PDE $F_v(u) = 0$ has unique solution given by $u = G(v)$ then solving the non-linear PDE is just solving the fixed-point problem $G(u)=u$.
Whether we establish the existence of solutions in this manner or another, the most useful application comes in analysing these solutions - since $u$ is a solution of the linear PDE $F_u(\cdot)=0$, you can apply all the same a priori estimates you're familiar with from the linear elliptic theory. All you have to do is control the properties of the solution enough for the operator $F_u$ to be elliptic.
In this case, you are given that $u$ is non-negative, so you should be able to verify that $F_u$ is indeed elliptic. Thus the maximum principle holds for $u$.
