  We consider the existence and uniqueness of a minimizer of the extremal
problem for weighted combined energy between two concentric annuli and obtain
that the extremal mapping is a certain radial mapping. Meanwhile, this in turn
implies a Nitsche type phenomenon and we get a
$\frac{1}{|w|^{\lambda}}-$Nitsche type inequality ($\lambda\neq1$). As an
application, on the basis of the relationship between weighted combined energy
and weighted combined distortion, we also investigate the extremal problem for
weighted combined distortion on annuli. This extends the result obtained by
Kalaj in \cite{Ka1}.
