  In this paper, we establish the first variational formula and its
Euler-Lagrange equation for the total $2p$-th mean curvature functional
$\mathcal {M}_{2p}$ of a submanifold $M^n$ in a general Riemannian manifold
$N^{n+m}$ for $p=0,1,...,[\frac{n}{2}]$. As an example, we prove that closed
complex submanifolds in complex projective spaces are critical points of the
functional $\mathcal {M}_{2p}$, called relatively $2p$-minimal submanifolds,
for all $p$. At last, we discuss the relations between relatively $2p$-minimal
submanifolds and austere submanifolds in real space forms, as well as a special
variational problem.
