Victoria Bampton can answer your question far better than I at this link-
Then- recheck your import options to be sure your 'Destination' folder is set to the external drive.
In the future make sure you have the external drive selected in the Destinations area of the import dialog window. Some times LR will default to the standard Default location for the destination. That would be in your User Name > Pictures folder.
Make sure the OS is seeing your external drive before you click the Import button. Then check the Destination area.
If you are transferring the photo files to the external drive (without using LR) and afterwards importing them, be sure that the import options at the top center of the Import page is set to Add and NOT Copy or Move.
I had this same problem: I was careful to make sure that LR was only going to catalog the image files, not copy them over, and yet it still filled up my hard drive.
Is there a cache that might fill up? I checked and the catalog file is only a couple of gigs in size.
Thanks for any info.
Are you importing via COPY or ADD?
What disk does Lightroom think the photos are located? Select a photo in Lightroom and then right-click on it (Windows) and select Show in Explorer; or Ctrl-clik on it (Mac) and select show in FInder.