On a class of special Euler-Lagrange equations

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