Sharpening Using Laplacian Kernel - How to Handle the Output (Specifically Negative Values)?
@SOFUser, Welcome to our community.

Since the High Pass Filter, such as the Laplacian Kernel, output has zero mean (As the DC Component is removed) it indeed can (Should) have negative values as the output oscillates around zero.

What to do with it depends on the objective:


Display Purpose
In case on wants to display the output of the Sharpening / HPF Filter process it is very common to add 50% gray to it. In case of 8 [Bit] image it means adding 127 / 128. This is what's don in Adobe Photoshop.
Processing Purpose
Usually we keep the negative values as they might be important in the next steps. For instance in Laplacian Pyramid if one won't keep the negative values the reconstruction won't be accurate. Another example would be sharpening where we add the output of the Laplacian Filter scaled by the sharpening factor to the image to have sharpened image.
Generating Image
If one wants to generate image and archive it in one of the UINT based formats (JPEG, PNG, etc...) usually one would clip data into the compatible range.