Reuse of a DH / ECDH public key
Using DH when both sides have static public keys (always send the same message when running the DH protocol) doesn't make DLP easier. It does increase the value of the secret key.

There are some things to worry about. If the DH shared secret or one of the secret keys is compromised, the compromise is complete. This is mitigated if the static public keys are replaced often.

In ordinary DH, the shared secret is a one-time secret. With static public keys, you have a multiple-use secret. You need to be very careful when using this key.

You should also be aware that DH where only one side has a static public key is essentially equivalent to ElGamal encryption...