  In this paper, we introduce the so-called $L_p$ $q$-torsional measure for
$p\in\mathbb{R}$ and $q>1$ by establishing the $L_p$ variational formula for
the $q$-torsional rigidity of convex bodies without smoothness conditions.
Moreover, we achieve the existence of solutions to the $L_p$ Minkowski problem
$w.r.t.$ the $q$-torsional rigidity for discrete measure and general measure
when $0<p<1$ and $q>1$.
