Adrenaline Bar is not trademarked. There in one Adrenaline Sports Bar in another state. We're struggling to understand at what point other fitness businesses have legal rights if their name incorporates the word Adrenaline but is a different combination and they have not trademarked it.
The crux of any trademark infringement issue is likelihood of confusion. Depending on what exactly these other fitness places are naming their "Adrenaline _______" business, your trademark might infringe it, as you will be offering the same exact service (fitness) and might confuse existing customers of the already existing "Adrenaline _______" business.
Consult with a trademark/business attorney. Many attorneys on this site, including myself, offer free consultations.
You should retain a trademark attorney who can analyze whether your proposed usage of the ADRENALINE BAR mark in Class 41 will infringe ADRENALINE SPORTS BAR in Class 43. This may also include a detailed search of both the USPTO trademark registry and State common law databases to determine if there are any other existing marks that may be confusingly similar. Although not registered, the owner of the Adrenaline Sports Bar in the other State may have acquired certain exclusive trademark rights in the ADRENALINE SPORTS BAR mark through the usage of the mark on a local or regional basis. Nevertheless, a qualified trademark attorney should be able to provide you proper guidance on whether your proposed mark is good or whether you should consider some alternatives. Good luck!
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Trademark Application Attorney
I would also advise you to retain an intellectual property attorney. As a general rule, the examiners at the USPTO are only going to search for marks that are registered or pending on the federal register.
However, the other business can assert its common law rights (ie it's been in business than longer than you) and try to cancel or oppose your trademark application or sue you for infringement. It is hard to predict if and when a business will do something like that--but it's always a risk. And certainly if you plan to expand into the state where the other Adrenaline Sports Bar is already in business, this is more likely to pose a risk for you going forward.
Patent Infringement Attorney
Short Answer: It depends.
"Common Law" Trademark rights accrue by creation and use of the mark. Even though the other business has not registered "Adrenaline" they may object to your attempt to register it. That does not mean that they will prevail, but it does mean that you need to hire an attorney that specializes in this area to consider and weigh your options. Good Luck.