  Let $A_1$ and $A_2$ be two circular annuli and let $\rho$ be a radial metric
defined in the annuli $A_2$. We study the existence and uniqueness of the
extremal problem for weighted combined energy between $A_1$ and $A_2$, and
obtain that the extremal mapping is a certain radial mapping. In fact, this
extremal mapping generalizes the $\rho-$harmonic mapping and satisfies equation
(2.7) obtained by mean of variation for weighted combined energy. Meanwhile, we
get a $\rho-$Nitsche type inequality. This extends the results of Kalaj (J.
Differential Equations, 268(2020)) and YTF (Arch. Math., 122(2024)), where they
considered the case $\rho=1$ and $\rho=\frac{1}{|h|^{2}}$, respectively.
  Moreover, in the course of proving the extremal problem for weighted combined
energy we also investigate the extremal problem for the weighted combined
distortion (see Theorem 4.1). This extends the result obtained by Kalaj (J.
London Math. Soc., 93(2016)).
