One considers an integral functional depending only on the trace of the metric tensor induced by a mapping of an n-dimensional domain Ω into ℝⁿ. One seeks a mapping which minimizes this functional. Under a well-defined smallness in the boundary conditions, one proves the existence of an infinite set of critical points of the functional. Under additional restrictions, one discusses an existence theorem and the character of the extremum. The convexity of the functional is not assumed. Such functionals are encountered in elasticity theory.