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?