I applied for PhD position and later I received an email telling that: I regret to inform that you have not been selected for this position, how may I send a follow up email, if should I do so?
In my opinion, there's really no need to respond to this. The sender was in all likelihood support staff and so won't forward your response further. But if you feel compelled anyway, you can write a very short message along the lines of: "Thank you very much for your consideration." Perhaps you would like some explanation, too. (I found their message to you rather terse!). In that case you may add: "I'd appreciate any insight into the Committee's decision not to pursue my candidacy for this position."
Dear Sir or Madam,
Thank you very much for your consideration. While it's regrettable that the decision was not made to support my candidacy, I would nonetheless appreciate any insight into the Committee's decision not to pursue my application further so that I can improve the competitiveness of my profile for future applications. If possible, could you kindly provide some input on that issue? Thanks again.