On the validity of the maximum principle and of the Euler-Lagrange equation for a minimum problem depending on the gradient

We consider the limiting case alpha = infinity of the problem of minimizing integral(Omega) (\\del u(x)\\(alpha) + g(u))dx on u is an element of + u(0) + W-0(1, alpha) (Omega); where g is differentiable and strictly monotone. If this infimum is finite, it is evidently attained; we show that any minimizing function u satisfies the appropriate form of the Euler-Lagrange equation, i.e., for some function p, div p(x) = g'(u(x)) for p(x) is an element of partial derivative(jB)(del(x)); where j(B) is the indicator function of the closed unit ball in the Euclidean norm of R-N and partial derivative is the subdifferential of the convex function j(B).