In this kind of fanfiction scenario, Snape might initially be conflicted. On one hand, he has his long - standing dislike for James Potter, Harry's father. But on the other, his sense of duty as a Hogwarts professor could kick in. He might reluctantly start to heal Harry, all the while muttering under his breath about how much trouble Harry always seems to get into, just like his father. Then, as he heals Harry, he might start to see some of Lily, Harry's mother, in him and soften a bit.
One place to start looking for the missing fanfiction could be on large fanfiction platforms like FanFiction.net or Archive of Our Own. They have extensive collections of Harry Potter fanfictions. Check their search functions, maybe filter by characters (Snape and Harry Potter) and relevant tags.
If Snape finds Harry Potter fanfiction, he'd probably be quite irritated. He has a rather stern and no - nonsense personality. He might see it as a frivolous waste of time and be disdainful of the creative liberties often taken in fanfiction. He'd likely scoff at the idea of people writing about his students and the wizarding world in such unregulated ways.
Snape would likely be angry at first. Harry has always been a source of trouble in his eyes, and running away would just add to that perception. He might mutter some sarcastic remarks under his breath about Potter's irresponsibility.
Look for any unique plot details you remember. If it involves Snape finding Harry Potter in a particular place, like a hidden chamber or a different magical realm, use that as a search keyword. For example, if it was in a 'forbidden forest' setting, search for 'Snape finds Harry Potter in the Forbidden Forest fanfiction'.
Snape might suspect that there is some dark magic at play. Given his knowledge and expertise in the Dark Arts, he would immediately look into any signs of foul play. He could also be conflicted because on one hand, Harry is a student he has had many issues with, but on the other hand, he is also an important figure in the fight against Voldemort. So he would have to decide whether to report Harry's disappearance or try to handle it privately.
He would probably be in shock. Snape has spent so much time around Harry in one way or another throughout his years at Hogwarts. His immediate reaction could be disbelief. He might stand there frozen for a moment, unable to process what he's seeing. Then, the full weight of the situation would hit him, and he could break down in a rare display of raw emotion.
He could be secretly worried. Snape might use his knowledge of potions and dark magic to try and find a cure or at least a way to stop the bleeding. He may also be conflicted, as his past relationship with Harry and his loyalty to different sides in the wizarding world play a role in how much he wants to help Harry.
In the fanfiction world, if Snape finds Harry cutting, he might be initially shocked and angry. He could see it as a sign of Harry's inner turmoil. Snape, despite his complex relationship with Harry, might feel a sense of responsibility. He might immediately take steps to stop Harry from further harming himself, perhaps using his magic to heal the cuts. And then he would likely question Harry about why he was doing such a thing, trying to get to the root of the problem.
In some fanfictions, Snape might use his advanced knowledge of potions to create a special concoction to heal Harry. He could draw on his skills in Herbology as well to find rare ingredients for the potion.