1. The Elbrus Region is close to the border between Russia and Georgia. Obtaining a special border permit is required to enter most of the gorges on the right side of the Baksan Valley. As an application for the border permit has to be made now (early 2018) at least 30 days before your actual arrival date you can do this through an agent only. There is no official fee for issuing the permit but the process takes time and some energy.
2. Prielbrusie National Park permits. In 2017 the costs were like this: 50 roubles per day spent in the National Park territory and 1200 roubles per person for being on the slopes of Mt Elbrus. The latter sometimes is called the Elbrus climbing permit. The National Park rangers check the permit at Azau and Krugozor ski lift stations. You can meet them on Mount Elbrus slopes as well - usually up to 3900m. If you have not obtained the permit in advance you can buy it from the rangers.
The National Park office is located in the town of Elbrus in the Baksan Valley. The system may change in 2018 season.