The paper concerns the following minimization problem \[ \text{Min } \int_\Omega [f(\|\nabla u(x)\|)+ g(x,u(x))] dx,\quad u- u^0\in W^0\in W^{1,1}_0(\Omega), \] where \(\Omega\) is an open bounded domain with Lipschitz boundary, \(W^0\) is a linear subset of \(W^{1,1}_0(\Omega)\) which contains all \(w\in W^{1,1}\) with \(\int_\Omega f(\|\nabla w(x)\|) dx< \infty\) and such that \(w\) can be suitably approximated by \(C^1\) maps. The author proves the validity of an Euler-Lagrange equation for this minimization problem under growth assumptions on \(f\) which are very general. In particular, this result contains the cases \[ f(\|\xi\|)= p(\|\xi\|) e^{K\|\xi\|}\quad(K\geq 0,\;p\text{ a polynomial}), \] \[ f(\|\xi\|)= \|\xi\|^{\|\xi\|}. \]