On the validity of the Euler–Lagrange equation in a nonlinear case

We prove the validity of the Euler–Lagrange equation for the class of functionals of the type ∫Ω[L(∇u)+g(x,u)]dx, where u↦g(x,u)u↦g(x,u) is a concave function. Our result is based on a recent paper by Degiovanni and Marzocchi (2009) [6], where no growth assumptions on LL are needed.