  Let $n\ge 2$ be an integer and assume that
$\mathbb{A}=\{x\in\mathbf{R}^n:1<|x|<R\}$ and $\A_\ast = \{y \in \mathbf{R}^n:
1 < |y| < R_\ast\}$ be two annuli in Euclidean space $\mathbf{R}^n$. Assume
that $\mathcal{F}(\A, \A_\ast)$ (resp. $\mathcal{R}(\A, \A_\ast)$) be the class
of all orientation preserving (resp. radial) homeomorphisms $h : \A \mapsto
\A_\ast$ in the Sobolev space $\mathcal{W} ^{1,n}(\A, \A_\ast)$ which keep the
boundary circles in the same order. In this paper, we extended the
corresponding results of Iwaniec and Onninen which was published in {\it Math.
Ann.} Vol. 348, 2010.
