Applying for PhD programs abroad while doing one in your home country In September I am starting the second year of PhD in my home country. At the same time, I am also going to apply for foreign PhD programs in the same science this year. It has always been my intent to apply for PhD in a prestigious American or European university. The reason I didn’t do it one year ago is that I wanted to gain some more working experience and knowledge before leaving to work abroad. Now I feel that I’m ready. What is more, now I have publications and good CV with talks at conferences, short research visits and such things. 

I would like to know, what is the general point of view on such PhD applicants as me? Should I mention in my applications that I want to do double PhD or maybe not even mention this at all? 

I should say that applying abroad, I don’t intend to leave the PhD program I’m currently doing. I know many students who obtained their master’s degrees at my university and then applied for PhDs abroad and left to other countries and at the same time were accepted to continue our PhD program. Now they are doing double PhDs, you could say. Of course, their situation was slightly different at the moment of applying, as the universities that accepted them didn’t know if they were actually interested in doing double PhD. However, practically it doesn’t seem to make any difference now.
It's usually a bad idea to do two PhDs at the same time because:


A PhD is a full-time job, with some very intensive periods. One has to deal with enough stress already when doing one PhD.
It would unavoidably generate conflicts: sometimes your advisor no 1 will ask you to do something which will keep you busy the full week or month. What do you tell your advisor number 2, that you're on holiday? If your two advisors don't agree on the direction of your research, what do you do? What if one insists to have their name on a paper you did with the other?
Either you do your best on both, but then it will take you more time than doing only one, probably double. Or you do one seriously and you botch the other, but then what's the point of doing it?
Assuming you survive the two PhDs, at the end you can apply for a postdoc or industry job... Exactly the same as with one PhD, no benefit.