You can get a trademark to protect against infringements on your domain name. It is essential for a online business. It is used for branded domain names. For an intellectual property as your domain name it is an intangible right, the benefit you'll get of ownership is that you the owner has a right of enforcement. So... yes you can do the trademark or do a intellectual property registration if you want to protect your domain name.
I own several trademarks that tie to my domains, as this allows me to register/own domains in other markets. That is for me the value of the trademark, as some countries won't let you own a domain unless you are a local company in that country or you have a international trademark for the brand/domain itself. Getting a trademark in the US costs less then $100 and can be done within a couple weeks, so really no reason not to do it. Just don't get into having to pay large amounts of money for this, as you don't need the most expensive TM company with the most costly lawyers to do it for you. Just need a credit card, about 1 hour of your time and you can register a trademark, any person or company can own a trademark in the US. You don't need to be a US person or US company.
I am not sure what exactly you are asking, as your question is not complete and you offered no additional detail.
But, what I think you are asking: Do you need to file a trademark for a domain name.
In the US, the answer is no. You *can* apply for a trademark for your brand name which may also be your domain name. But a trademark is not required for a domain.
On the flip side, if someone owns the trademark to the word/words of your domain, they may pursue legal options to prevent you from using said domain name.
All the best,