Variational problems in $L^{\mathrm{\infty }}$ involving semilinear second order differential operators

For an elliptic, semilinear differential operator of the form $S(u)=A:D^{2}u+b(x,u,Du)$, consider the functional $E_{\mathrm{\infty }}(u)={}_{\mathrm{\Omega }}\mathrm{\mid }S(u)\mathrm{\mid }$. We study minimisers of $E_{\mathrm{\infty }}$ for prescribed boundary data. Because the functional is not differentiable, this problem does not give rise to a conventional Euler-Lagrange equation. Under certain conditions, we can nevertheless give a system of partial differential equations that all minimisers must satisfy. Moreover, the condition is equivalent to a weaker version of the variational problem.