What settings do you have in the Sharpening and Image Sizing sections of the export dialog box?
What is the size of the (cropped) original before export — in pixels, height and width?
What software are you using outside of Lightroom that shows the export is not sharp?
Can we see 1:1 screen captures of the photo in the Develop Module and then after export?
Well, since you don't think there is anything wrong with your export settings, its hard to see how we can help.
What happens if you zoom in on your photo in the Library Module and then unzoom? Does it get sharper?
Well, @wertvoll, I did ask for other pieces of information, as well as 1:1 screen capture crops of Develop Module and Exported Photo.
Without this information, its hard to advise any further.
The screenshot you’ve posted appears to be Fit zoom, rather than 1:1 zoom, and it should look different between the two modules due to how the reduced-sized screen representation is computed--the resampling algorithms are different between Develop and Library with Develop's being less accurate for speed reasons, and usually causes the Fit zoom in Develop to look artificially sharp. Library previews are computed in the background and therefore can afford to take more time. Develop previews are computed on the fly as you move each slider so cannot take very long at all.
Can you show us how things look at 1:1 zoom in Library and Develop and an exported file, with the zoom slider in the screenshot of those from LR? If necessary, recompute the Library 1:1 preview for the image you’ve chosen to show.
Thanks for helping out! In 1:1 zoom it looks the same. Nevertheless the Library Module and export don't consider my adjustment brush in the Develop Module. It just ignores my editing and exports what I don't want.
“In 1:1 zoom it looks the same” means what, exactly? That the Library and Develop images look the same as each other or that at 1:1 zoom the Library image is less sharp than the Develop image the same as when you use Fit zoom?
There was a problem with early versions of LR 5 where Develop and Export had different amounts of sharpening, basically where sharpening and noise-reduction were ignored if the image was resized to less than 1/3 of the original size. But that situation has been fixed some versions ago, already, as far as anyone has been able to demonstrate on here.
If you are able to demonstrate that 1:1 zoom Library and Develop look different then you may have found a bug, but so far despite many initial reports, no one has demonstrated the issue. They usually stop posting when asked for proof at 1:1 zoom, and it seems like that may be occurring here.
You are saying that Library and Export are ignoring your brush edits. What happens if your brush includes a large increase or decrease in Exposure—you can change an existing brush stroke set temporarily to test—does that adjustment also get ignored, or only adjustments related to sharpening and noise-reduction?
The only issue I know of where Library and Develop look a bit different is when the images are very high-ISO so that they are full of tiny noise specs, and those do resize down differently, but I’m guessing that is unlikely to be the issue in your portrait since it seems to be more formal, perhaps with studio lighting or a tripod and slow shutter speed so high-ISO wasn’t necessary. Of course this is just a guess based on looking at a small portion of the photo, above.
The bottom line is that detail adjustments like Sharpening and Noise-Reduction need to be judged at 1:1 zoom, not Fit or other smaller-than-1:1 zoom, otherwise what you see in Develop is almost guaranteed not to translate to Export or Library.