On the validity of the Euler-Lagrange equation (Q5937333)

The paper concerns the following minimization problem NEWLINE\[NEWLINE\text{Min } \int_\Omega [f(\|\nabla u(x)\|)+ g(x,u(x))] dx,\quad u- u^0\in W^0\in W^{1,1}_0(\Omega),NEWLINE\]NEWLINE 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.NEWLINENEWLINENEWLINEThe 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 NEWLINE\[NEWLINEf(\|\xi\|)= p(\|\xi\|) e^{K\|\xi\|}\quad(K\geq 0,\;p\text{ a polynomial}),NEWLINE\]NEWLINE NEWLINE\[NEWLINEf(\|\xi\|)= \|\xi\|^{\|\xi\|}.NEWLINE\]