Checking whether a number exists or not would not be a valid way of verifying the number. For instance, my former number exists but it is not in service. Even so, I do not believe there is one central way to verify that it "exists". However, if you are looking to be sure that the number put into the box is the customers' number, there are a couple ways of doing this.
The first way would be to use a service to verify the number for you. This is one example of that type of service (not a recommendation, just an example):
Another way that would be cheaper would be to do the verification yourself. If you are only verifying mobile phones, you could send a text to the phone with a generated number (just a short random string). Have the user type that string into your form to verify they were the ones that got the message. To do the actual sending of the text message, you could go cheap and email the text message email address of the phone (different for each carrier) or you could use a service such as Twilio
] to send the message (at one cent per text inside the US). I do recommend Twilio. I've used them in the past and they are excellent, both in terms of their service as well as their customer support.