Are there other governments in The Hunger Games world?

We know that Panem exists in a post-apocalyptic North America but are there other established governments on other continents?

Answer

None that we know of. Panem is, as far as anyone else in the story knows, the last human civilization on Earth. The events preceding the creation of Panem (a combination of natural disasters and man-made war and disease), some 200 years ago, have rendered most of the rest of the world, including most of North America itself, uninhabitable, and dangerously depleted the world's population.

Now, from a purely realistic perspective, it seems unlikely that Panem's people are the only humans left on Earth. There are plenty of people living in plenty of places that would neither be very susceptible to a major natural disaster, nor worth the effort of deploying a WMD. So, there is every chance of the other major landmasses (Africa, Eurasia, South America and Australia) harbouring some remaining human populations, probably mainly nomads moving from place to place hunting what animals are left.

However, those countries and their people are few and far between. Mongolia, for instance, is not real high on anyone's target list, but it has the bad fortune of being landlocked in between two of the world's foremost military and nuclear powers. Similarly, the Himalayan nations are no real threat to anyone, but lie in the middle of a trifecta of conflict (China, India, Pakistan). Pretty much every nation in Europe, South and Central America, the Middle East, and Africa are on SOMEONE's hit list. Australia is too close to Western power factions to escape unscathed, and the most habitable areas are the ones with the major cities (and close to the ocean, meaning a rise in sea levels would submerge a large part of the habitable land).

North America is of course on many hit lists; the primary reason any of it would survive is probably due to the United States having invested relatively early in a missile defence system. That at least prevents or reduces the nuclear attacks from sovereign nations. We'd still be subject to global warming, epidemic, and terrorist attack rendering large swaths of the most densely-populated areas inhospitable. What's left are small tracts of land in between what used to be dense population centres, that have some remaining economic value: farming (11), ranching (10), fishing (4), mining (12 & 2), textiles (8), lumber (7), and what's left of the tech and consumer goods industries (1,3,9).

But, the basic idea is that while the Capitol has advanced technology significantly in many areas, in others they are woefully limited compared to today. For instance, they have "hovercraft", the exact nature of which is not detailed in the books but is probably based on ducted-fan propulsion much like the craft in Avatar. These craft are short-to-medium range, probably using a battery charged at the home base. Fossil fuels of many kinds are in such short supply that air planes as we know them today are infeasible (though not unknown to the more educated people of Panem). So too are large ocean ships. So, trans-oceanic travel is generally considered impossible by Panem's residents (none of which would have any knowledge of sailing ships), and thus even if people were still scraping out a living elsewhere on the planet, Panem has neither the will nor the resources to go find out.