Variants of Bilinear Diffie-Hellman Assumption
To the best of my knowledge, this hard assumption is introduced by Boneh and Boyen in this paper. 

But I don't think so the assumption that you mention being hard, because $c=0$ is a simple solution for it. Then the element $g^{\frac{1}{s}}$ should not publish. Also, this assumption is not bilinear because the challenge is an element in the cyclic group of $\mathbb{G}$ and it is not $e(g,g)^{\frac{1}{\alpha+c}}$.