  4-harmonic and ES-4-harmonic maps are two generalizations of the well-studied
harmonic map equation which are both given by a nonlinear elliptic partial
differential equation of order eight. Due to the large number of derivatives it
is very difficult to find any difference in the qualitative behavior of these
two variational problems. In this article we prove that finite energy solutions
of both 4-harmonic and ES-4-harmonic maps from Euclidean space must be trivial.
However, the energy that we require to be finite is different for 4-harmonic
and ES-4-harmonic maps pointing out a first difference between these two
variational problems.
