Public key encryption with equality test (PKEET) can check whether two ciphertexts are encrypted from the same message or not without decryption. This attribute enables PKEET to be increasingly utilized in cloud storage, where users store their encrypted data on the cloud. In traditional PKEET, the tester is authorized by the data receiver to perform equality test on its ciphertexts. However, the tester can only test one ciphertext or all ciphertexts of one receiver with one authorization. It means that the receiver cannot adaptively authorize the test right of any number of ciphertexts to the tester. A trivial solution is authorizing one ciphertext each time and repeating multiple times. The corresponding size of trapdoor in this method is linear with the number of authorized ciphertexts. This will incur storage burden for the tester. To solve the aforementioned problem, we propose the concept of PKEET supporting partial authentication (PKEET-PA). We then instantiate the concept to a lightweight PKEET-PA, which achieves constant-size trapdoor. Besides, we prove the security of our PKEET-PA scheme against two types of adversaries. Compared with other PKEET schemes that can be used in trivial solution, our PKEET-PA is more efficient in receivers' computation and has lower trapdoor size.