I am looking for an example of maximizing a functional with dependencies on 0th and 1st order derivatives for a probability cumulative distribution function or density function, preferably under some additional integral constraint. Hopefully this would involve the use of Euler Lagrange equation with Lagrange multiplier. The problem I am facing is that one cannot in general assume smoothness or even continuity of the resulting solution, but I am not sure what kind of pathology could arise as a result.
All the elementary examples given in the literature for Euler Lagrange equation deal with geometric problems like minimizing the arc length of curves or isoperimetric inequality types of result. For instance, here is such a lecture note.