A simple way would be through their powers. Harry sees Percy's instant and powerful control over water, while his own magic related to water is much more limited. This makes him envious.
The fanfic could show it in the way they are received by others. Percy is seen as a hero in his world and is adored by many. Harry, on the other hand, has faced a lot of skepticism at Hogwarts at times. This contrast could make Harry jealous of Percy's popularity and respect.
It could be shown in their relationships with their friends. Percy has this tight - knit group of friends from Camp Half - Blood who are always there for him. Harry has had his fair share of issues with friends like Ron and Hermione at times. Seeing Percy's harmonious friendships could make Harry jealous and the fanfic could explore how he copes with that.
Well, one possible scenario in such a fanfic could be that Harry Potter is jealous of Percy Jackson's connection to the sea. Since Harry has his own element of water in the form of the Black Lake at Hogwarts, but Percy has this vast and powerful connection to all of the oceans. It could create a really interesting dynamic where Harry feels overshadowed in a way.
In a 'Percy Jackson adopts Harry Potter' fanfic, the plot could start with Percy being on some kind of mission that crosses into the wizarding world. He stumbles upon Harry, who is alone and vulnerable. Percy decides to adopt Harry because he sees the same kind of determination and courage in Harry that he has. As they journey together, they exchange knowledge. Percy teaches Harry about Greek mythology and combat skills, while Harry shares his knowledge of magic. They form a bond that helps them face threats from both magical and mythological creatures, and they might even team up to fight against a common enemy that is a threat to both their worlds.
One way it could be possible is through some sort of magical multiverse concept. Maybe there are different dimensions where the magic systems of both worlds collide. For example, a powerful magic spell or a strange rift in time and space could bring them together as brothers. It would be really interesting to see how Harry's wizarding skills and Percy's demigod powers interact in such a scenario.
Percy could cope by relying on his friends. His friends are very resourceful and they would surely help him out. They could carry him around when needed and be his eyes and ears in dangerous situations.
One great fanfic could be 'Jealousy in Hogwarts Halls'. In it, Harry's jealousy of Percy might stem from Percy's academic achievements and the way he's always so well - regarded by the teachers. It shows how Harry, who already has so much on his plate, struggles with this new emotion. It's a really interesting take on the characters and their relationships within Hogwarts.
Gosling could be a time - traveling character. He could travel from the real world into the Harry Potter universe and then find a way to enter the Percy Jackson world. His knowledge from the modern world could help him solve problems in both magical realms.
One possible interesting plot could be that the magic from the Harry Potter world starts to seep into the Percy Jackson universe, causing chaos among the demigods. For example, Hogwarts could suddenly appear in the middle of Camp Half - Blood. The demigods, not used to this type of magic, would be both curious and threatened. Meanwhile, Harry and his friends could get accidentally transported to the Percy Jackson world and have to team up with the demigods to figure out how to send Hogwarts back and restore the balance between the two magical systems.
A possible plot is that Voldemort discovers the existence of the demigods and tries to use their powers for his own evil purposes. He sends his Death Eaters to capture some demigods. Percy and his friends manage to escape and seek refuge at Hogwarts. At Hogwarts, they learn about magic in a whole new way while also teaching the wizards and witches about their own powers. The two groups then have to combine their skills, like the demigods' combat abilities and the wizards' spell - casting, to fight against Voldemort and his new - found allies from the world of Greek mythology.
In some fanfics, they might be depicted as brothers in some sort of crossover scenario. Fanfic writers often like to create unique relationships between different fictional universes' characters.
There's a fanfic called 'The Quest Beyond Worlds'. It takes Percy and Harry on a cross - universe adventure. It delves deep into the mythology of both series, making it a very engaging read. It also has some great character development for both Percy and Harry as they learn from each other's worlds.