There is currently no mobile application dedicated exclusively to PS Now. In the official PlayStation app, you can click on a link that takes you to the PS Now page. Here you can explore the games that are currently available with a subscription. We noticed a strange feature. Even if your PS Now subscription is activated and recognized in the app, it will be displayed as the total price when you search for games.
When you search for a game in the PS Store on a PS5 console, it will tell you if that game is in a PS Now or PS Plus subscription. It is a pity that the mobile app does not notify you of the games included in your membership once you have paid for them. Hopefully, Sony will incorporate these improvements shortly.