I have applied for a PhD program in USA. I have mentioned a professor's name in the SOP. Will the professor only get the application to review or the other professors will also be able to see my application? Will the other professors mind as I did not mention their name?
Usually PhD admissions in the US are by a committee of several professors. Everyone on the committee will see your application (except that applications are often triaged or only read in full by certain members); the professor you mentioned may or may not be on the committee.
If your application suggests you are interested in working with a particular professor, then your odds of admission go up if that person is taking students and wants to advise you - one of the things an admissions committee is looking for is "fit", and "fit" is pretty good in this circumstance.
Your odds of admission go way down if that person is not taking students or does not want to advise you. Again, the admissions committee will be looking for "fit" and this would suggest you would not "fit".
I wouldn't expect other professors to be offended, but it's quite reasonable to not admit someone who will not have a place. Typically I don't think it's a good idea to mention an individual professor unless you have had previous positive interactions with them and are applying to the program specifically to work with that person.