TITLE: References for existence of solutions to overdetermined system of partial differential inequalities
QUESTION [7 upvotes]: I want to show the existence of solutions to an overdetermined system of second-order partial differential inequalities in a given region $\Omega\subset\mathbb{R}^2$, 
\begin{equation*}
\begin{cases}
P_1(u)\geq 0,&\\
P_2(u)\geq 0,&\\
P_3(u)\geq 0.&
\end{cases}
\end{equation*}
Updated: $P_i$ are linear/quasilinear/nonlinear second-order differential operators, 
\begin{equation*}
\begin{cases}
P_1(u)=&\hspace{-0.3cm}xu_{xx}+yu_{xy}+x^2u_{yy}+4u_x.\\
P_2(u)=&\hspace{-0.3cm}u_{xx}u_{yy}-u_{xy}^2+(xu_x-yu_y)u_{xx}+(yu_x-x^2u_y)u_{xy}+x(xu_x-yu_y)u_{yy}.\\
P_3(u)=&\hspace{-0.3cm}(u_{xx}u_{yy}-u_{xy}^2)u_x+(xu_x^2-yu_xu_y-x^2u_y^2)u_{xx}+(yu_x^2-x^2u_xu_y-xyu_y^2)u_{xy}+(x^2u_x^2-xyu_xu_y-y^2u_y^2)u_{yy}.
\end{cases}
\end{equation*}
To do this, I am trying to solve for $(u,f_1,f_2,f_3)$ of the following system of PDEs,
\begin{equation*}
\begin{cases}
P_1(u)=f_1^2,&\\
P_2(u)=f_2^2,&\\
P_3(u)=f_3^2.&
\end{cases}
\end{equation*}
Could anyone suggest any references to deal with either system? 
Any suggestion is appreciated.
Updated: Is it true that if I find all compatible conditions for $f_1,f_2,f_3$ and they are satisfied, then this system has a solution?

REPLY [3 votes]: The Cartan--Kaehler theorem gives sufficient conditions for the existence of local solutions of any real analytic system of partial differential equations. However, the solutions are only local. See Bryant, Chern, Gardiner, Goldschmidt and Griffiths, Exterior Differential Systems, MSRI.