The validity of the Euler–Lagrange equation for solutions to variational problems

We prove higher integrability properties of solutions to the problem of minimizing $\int_{\Omega}L(x,u(x),\nabla u(x))\rm{ d}x,$ where $\xi\mapsto L(x,u,\xi)$ is a convex function satisfying some additional conditions. As an application, we prove the validity of the Euler-Lagrange equation for a class of functionals with growth faster than exponential.