  We study the problem of minimizing the functional $$
I(\varphi)=\int\limits_{\Omega} W(x,D\varphi)\,dx $$ on a new class of
mappings. We relax summability conditions for admissible deformations to
$\varphi\in W^1_n(\Omega)$ and growth conditions on the integrand $W(x,F)$. To
compensate for that, we impose the finite distortion condition and the
condition $\frac{|D\varphi(x)|^n}{J(x,\varphi)} \leq M(x) \in L_{s}(\Omega)$,
$s>n-1$, on the characteristic of distortion. On assuming that the integrand
$W(x,F)$ is polyconvex and coercive, we obtain an~existence theorem for the
problem of minimizing the functional $I(\varphi)$ on a new family of admissible
deformations. KEYWORDS: functional minimization problem, nonlinear elasticity,
mapping with finite distortion, polyconvexity.
