Recently, many existence results for the stochastic thin-film equation were established in the case of a quadratic mobility exponent \(n=2\) , in which the noise term \(\partial _x\big (u^\frac{n}{2}\mathcal {W}\big )\) becomes linear. In the case of a non-quadratic mobility exponent, results are only available in the situation that \(n\ge \frac{8}{3}\) leaving the interval of mobility exponents \(n\in \big (2,\frac{8}{3}\big )\) untreated. In this article we resolve the current gap in the literature by presenting a proof, which works under the assumption \(n\in (2,3)\) , i.e., the regime of weak slippage. The key idea is to use that the \(\log \) -entropy dissipation coincides with the energy production due to the noise. To realize this idea, we approximate the stochastic thin-film equation by stochastic thin-film equations with inhomogeneous mobility functions, which behave like a higher power near 0. As a consequence the approximate solutions are non-negative, which is vital to use the \(\log \) -entropy estimate.
