Critical points of solutions of degenerate elliptic equations in the plane

We study the minimizer $u$ of a convex functional in the plane which is not G\^ateaux-differentiable. Namely, we show that the set of critical points of any $C\sp1$-smooth minimizer can not have isolated points. Also, by means of some appropriate approximating scheme and viscosity solutions, we determine an Euler-Lagrange equation that $u$ must satisfy. By applying the same approximating scheme, we can pair $u$ with a function $v$ which may be regarded as the stream function of $u$ in a suitable generalized sense.