Does there exist a Laplace transform of $W(at)$ that can be expressed in terms of elementary functions and the Lambert W function $W(x)$?
If such a transform exists, how is it expressed in terms of $s$, and how is it derived?
A couple properties of the productlog $W(x)$ to note include: $$ e^{W(x)}=\frac{x}{W(x)}\\ W'(x)=\frac{W(x)}{x\left(1+W(x)\right)}$$ and $$\int W(ax)=x(W(ax)−1)+\frac{1}{a}e^{W(ax)}+C.$$
On
Laplace transform LW(s) of Lambert function W(t) is
$$\frac {1} {2i} \int_{c-i\infty}^{c+i\infty}s^{-1-y}\frac {y^{y-1}} {sin\ {\pi y}}\,dy,{\,c \in (0,1)}$$
which is Mellin-Barnes integral. This integral converges for all complex s excluding singularities, of course. I cannot express this integral by known "named" functions.
Other possibility is to calculate integral
$$ \frac {1} {s} \int_{0}^{\infty} e^{-u} W(\frac {u} {s})\,du$$
i.e. we integrate complex valued function along real u > 0. This integral converges for all complex s excluding singularities as well.
I propose to define a new special function, namely LW$(x)$ : $$\text{LW}(x)=\int_0^\infty W(t)e^{-x\:t}dt$$ where W is the Lamber's W function.
In the futur, if this brand new function becomes standard, if it acquires the honorific status of standard special function, if it spread in the literature with a lot of studies of properties, if it becomes familiar, if it is implemented in mathematical softwares, then you could say :
"The Laplace transform of $\quad \text{W}(ax)\quad$ is $\quad\frac{1}{a}\text{LW}(\frac{s}{a})$."
This would be a typical case of special function emergence, exactly as many special functions emerged : https://fr.scribd.com/doc/14623310/Safari-on-the-country-of-the-Special-Functions-Safari-au-pays-des-fonctions-speciales