There are many ways that an OS can handle the scenario described in the question, which is how to effectively deal with running out of free RAM. Depending on the CPU architecture and the goals of the OS here are some ways of handling this issue.
One solution is to simply kill processes when they attempt to malloc(or some other similar mechanism) and there is no free pages available. This effectively avoids the problem posed in the original question. On the surface this seems like a bad idea, but it has the advantages of simplifying kernel code and potentially speeding up context switches. In fact, in some applications if the code running had to use swap space to accommodate pages that can not fit into RAM by using non-volatile storage, the application would take such a performance hit that the system effectively failed anyways. Alternatively, not all computers even have non-volatile storage to use for swap space!
As already alluded to, the alternative is to use non-volatile storage to hold pages that can not fit into RAM. Actual specific implementations can vary depending on the specific needs of the system. Here are some possible ways to directly answer how the mappings of V1->P1 and V2->P1 can exist.
1 -There is often no strict requirement for the OS needs to maintain a V1->P1 and V2->P1 mapping. So long as the contents of the virtual space stays the same, the physical address backing it is transparent to the program running. If both programs needed to run concurrently the OS can stop the program running in V2 and move the memory of P1 to a new region, say P2. Then remap V2 to P2 and resume running the program in V2. This assumes free RAM exists to map of course.
2 - The OS can simply choose not to map the full virtual address space of a program into a RAM backed physical address space. Suppose not all of V1 address space was directly mapped into physical memory. When the program in V1 hits an unpagged section the OS can catch the exception triggered by this. If available RAM is running low, the OS can then use the swap space in non-volatile storage. The OS can free up some RAM by pushing some physical addresses of a a region not currently in use to swap space in non-volatile storage (such as the P1 space). Next the OS can load the requested page into the freed up RAM, setup a virtual to physical mapping and then return execution to the program in V1.
The advantage of this approach is that the OS can allocate more memory then it has RAM. Additionally, in many situations programs tend to repeatedly access a small area of memory. As a result, not having the entire virtual address region page into RAM may not incur that big of a performance penalty. The main downside to this approach is that it is more complex to code, can make context switches slower and accessing non-volatile storage is extremely slow compared to RAM.