Yes, the world will adapt to overpopulation. Each time an overpopulation "milestone" is reached, science discovers a new technology to deal with the issue. In addition, overpopulation is only a problem in ill-educated countries, and as such, educating these countries in the use of birth control will make overpopultation a non-issue.
The earth and humans alike are constantly adapting to different situations we face. As we evolve and grow and learn more and more about this earth we are able to find new ways to deal with past problems. I believe we will likely adapt to an overpopulation as well and find a way to deal with it.
I's not just the number of people, it's the waste they produce and the damage to the earth that is done to support them. The earth if left to itself is a well buffered system, but as anyone who has taken junior high school chemistry knows, if you add enough of a negative solution, buffers can be overwhelmed.
The world is growing quicker than we can control it. If you look at the worlds population rate, it has skyrocket in such a short time that if the current rate of population continues and if tech progresses at it's current rate, we will run out of space on earth before we are ready to move to another planet. Also, due to our nature, we will not be able to work together and share the resources mandatory to help our insanely overpopulated world. We are on track to chaos, and if we don't do something, we will all suffer.