This problem isn't JUST the Optimus F6, it's a lot of Android devices.
If you're not technically proficient, such a problem may be insurmountable depending on how you use the phone.
The simple, built in, stock approach is to look in settings / storage - apps.
There, it lists each app where you can select them, one at a time, and select the "Move to SD card" option. This relocates much, but not all, of the application - and it helps, but there is always a point where 'stuff' accumulates, and you're back to the out of space problem.
If you don't have an SD card, you're basically left with one option - backup the APK's (maybe the data, too) and remove stuff from the phone.
Even when you have an SD card, it's not like a camera. You don't get extra space for applications and general operations simply because you mount the card.
I know everyone generally expects that to happen, AND IT SHOULD (!!), but this is a basic design flaw of Android installations in most (or all) devices.
The only way to deal with the problem is reorganization of the device, starting with rooting the device.
Link2SD can then be used, but it's like a grand version of the "move to SD card" - you'll still eventually run out of room, with plenty of spare on the SD card.
What we need is an ability inherent in Linux, which is the underlying operating system here, but is given odd twists in the Android builds everywhere. We need to be able to install "drives" ( the SD card ) and move partitions around, like one would do with a desktop PC.
You know full well, I'm sure, that if you had this problem on a PC (linux based, Windows based, MAC, all of them), you could add a drive and start putting anything and everything on it.
You'd think there's an easy way of doing that on Android. I have 32 years of experience in software engineering. I know Unix. I know Linux. I also know Android.
There is no easy way of doing it.
Yet.
I myself am still working on a really good way of fixing my F6 storage problem, but I would only offer suggestions about it to the technically savvy. Anyone not able to root the device (as it IS simple when you know how), should avoid the solutions I'm coming up with.
I'm still working out the details. It could be a while. I'm not sure if I can offer the suggestions here or not, but I'll think abou it.
The basic idea I'm working on involves creating filesystems on the SD which can be mounted at boot time instead of those built into the internal SD storage.
It is certainly possible to do.