  The study of higher order energy functionals was first proposed by Eells and
Sampson in 1965 and, later, by Eells and Lemaire in 1983. These functionals
provide a natural generalization of the classical energy functional. More
precisely, Eells and Sampson suggested the investigation of the so-called
$ES-r$-energy functionals $ E_r^{ES}(\varphi)=(1/2)\int_{M}\,|(d^*+d)^r
(\varphi)|^2\,dV$, where $ \varphi:M \to N$ is a map between two Riemannian
manifolds. In the initial part of this paper we shall clarify some relevant
issues about the definition of an $ES-r$-harmonic map, i.e, a critical point of
$ E_r^{ES}(\varphi)$. That seems important to us because in the literature
other higher order energy functionals have been studied by several authors and
consequently some recent examples need to be discussed and extended: this shall
be done in the first two sections of this work, where we obtain the first
examples of proper critical points of $E_r^{ES}(\varphi)$ when $N={\mathbb
S}^m$ $(r \geq4,\, m\geq3)$, and we also prove some general facts which should
be useful for future developments of this subject. Next, we shall compute the
Euler-Lagrange system of equations for $E_r^{ES}(\varphi)$ for $r=4$. We shall
apply this result to the study of maps into space forms and to rotationally
symmetric maps: in particular, we shall focus on the study of various family of
conformal maps. In Section 4, we shall also show that, even if $2 r > \dim M$,
the functionals $ E_r^{ES}(\varphi)$ may not satisfy the classical Palais-Smale
Condition (C). In the final part of the paper we shall study the second
variation and compute index and nullity of some significant examples.
