  The Minkowski problem for torsional rigidity ($2$-torsional rigidity) was
firstly studied by Colesanti and Fimiani \cite{CA} using variational method.
Moreover, Hu, Liu and Ma \cite{HJ00} also studied this problem by method of
curvature flows and obtain the existence of smooth even solutions. In addition,
the Minkowski problem for $2$-torsional rigidity was also extended to $L_p$
version and Orlicz version.
  Recently, Hu and Zhang \cite{HJ2} introduced the concept of Orlicz mixed
$q$-torsional rigidity and obtained Orlicz $q$-torsional measure through the
variational method for $q>1$.Specially, they established the functional Orlicz
Brunn-Minkowski inequality and the functional Orlicz Minkowski inequality.
  Motivated by the forgoing works, we can propose the Orlicz Minkowski problem
for $q$-torsional rigidity, and then confirm the existence of smooth non-even
solutions to the Orlicz Minkowski problem for $q$-torsional rigidity with $q>1$
by method of a Gauss curvature flow.
