Sure I can just get an answer from wolfram alpha, but I want to know the steps involved.
I noticed the title equation while reading this: https://cs.uwaterloo.ca/research/tr/1993/03/W.pdf
The paper shows you how to get to the equation, and how its solvability for $α = 1$ implies solvability for all $α \neq 0$, but not how to actually solve it. So, can someone please guide me through the general solution method?
Introducing $y=-\nu\alpha x^\alpha$ one must solve $y\mathrm e^y=-\nu\alpha$ hence indeed Lambert W function is involved, since $$x=\left(\frac{W(-\nu\alpha)}{-\nu\alpha}\right)^{1/\alpha}.$$ Note: whether this formula can be useful at all is somewhat doubtful.