Earning on the Binance app without an initial investment can be challenging, but the following methods might help:

1. Referral Program: Invite friends to join Binance using your referral link. When they sign up and start trading, you can earn a percentage of their trading fees.

2. Airdrops: Look out for airdrop events where new cryptocurrencies are distributed for free, often to holders of certain existing cryptocurrencies. Binance occasionally participates in such initiatives.

3. Staking: If you already own some cryptocurrencies, you can stake them on Binance to earn rewards by holding specific coins, even if you acquired them through other means.

4. Promotions: Binance frequently organizes promotional events like trading competitions or task-based challenges where you can earn rewards.

5. Learn and Earn: Binance offers a program where users can watch educational videos about cryptocurrencies, complete quizzes, and receive small amounts of those cryptocurrencies as rewards.

Always conduct thorough research and understand the associated risks before participating in these activities!