  We make some remarks on the Euler-Lagrange equation of energy functional
$I(u)=\int_\Omega f(\det Du)\,dx,$ where $f\in C^1(\mathbb R).$ For certain
weak solutions $u$ we show that the function $f'(\det Du)$ must be a constant
over the domain $\Omega$ and thus, when $f$ is convex, all such solutions are
an energy minimizer of $I(u).$ However, other weak solutions exist such that
$f'(\det Du)$ is not constant on $\Omega.$ We also prove some results
concerning the homeomorphism solutions, non-quasimonotonicty, radial solutions,
and some special properties and questions in the 2-D cases.

Analysis of PDEs