  J. Eells and L. Lemaire introduced k-harmonic maps, and Wang Shaobo showed
the first variational formula. When, k=2, it is called biharmonic maps
(2-harmonic maps). There have been extensive studies in the area. In this
paper, we consider the relationship between biharmonic maps and k-harmonic
maps, and show non-existence theorem of 3-harmonic maps. We also give the
definition of k-harmonic submanifolds of Euclidean spaces, and study k-harmonic
curve in Euclidean spaces. Futhermore, we give a conjecture for k-harmonic
submanifolds of Euclidean spaces.
