I am looking for a utility function that will lead to the following demand
$$\alpha p^{\epsilon}$$.
I know it is most likely a CES utility since the elasticity of demand is constant and this is usually consequence of CES utility.
I also believe that the second good most likely was set as the numeare good or that the function must be a special case when the relative prices do not matter.
I am also little bit confused because of the positive relationship between price and demand I guess it must be a Giffen good.
I tried to solve several problems with different CES utilities but did not ever got this. I am not sure if there is any analytical way how to get from result like this to initial problem, I am just using trial and error, I would appreciate any help.