Variational problems in L∞ involving semilinear second order differential operators

For an elliptic, semilinear differential operator of the form S(u)=A:D2u+b(x,u,Du), consider the functional E∞(u)=esssupΩ|S(u)|. We study minimisers of E∞ 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.