# EDP with complementary terms

I am considering a problem of a two-dimensional ODE involving Karush, Kuhn and Tucker conditions on one of the unknowns. After a few algebraic manipulations, I end up having to solve the following PDEs: \begin{aligned}&Q(x_2,y_2)=\frac{\partial P(x_2,y_2)}{\partial x_2}y_2+\frac{\partial P(x_2,y_2)}{\partial y_2}(f_2(x_2,y_2)+\lambda)\\ &f_1(x_2,y_2)=\frac{\partial Q(x_2,y_2)}{\partial x_2}y_2+\frac{\partial Q(x_2,y_2)}{\partial y_2}(f_2(x_2,y_2)+\lambda)\end{aligned} with: $$x_2-d\leq 0 \quad;\quad \lambda\geq 0 \quad;\quad (x_2-d)\lambda=0$$ In these equations, $d$, $f_1(x_2,y_2)$, and $f_2(x_2,y_2)$ are known; $P(x_2,y_2)$, $Q(x_2,y_2)$, and $\lambda$ should be found. I am not detailing the boundary conditions. Have you ever seen PDEs of this type (\$\lambda is problematic)? If yes, are you aware of a numerical strategy that would solve them?

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