  Let $1 \leq m \leq n$ be two integers and $\Omega \Subset \C^n$ a bounded
$m$-hyperconvex domain in $\C^n$. Using a variational approach, we prove the
existence of the first eigenvalue and an associated eigenfunction which is
$m$-subharmonic with finite energy for general twisted complex Hessian
operators of order $m$. Under some extra assumption on the twist measure we
prove H\"older continuity of the corresponding eigenfunction. Moreover we give
applications to the solvability of more general degenerate complex Hessian
equations with the right hand side depending on the unknown function.
