gabrieletera before you do anything else, and if you like your phone then you want to stop using it right now. Stop syncing/charging it and stop turning it on. You have water damage and rice does not do anything.Rice is not a tool and using it, is the equivalent of not doing anything. Check on here for more on that.
What you do want to do is to properly clean your phone. Disassemble your phone using these guides . You must remove all EMI shields. If you do not remove those, you might as well not do anything. After you removed those, clean the whole board, connectors and all with +90% isopropyl alcohol. Follow this guide , and even so it was written for a 3G all the points are still pertinent to your phone. While you clean your board, check for any obvious damage, like burned or missing components etc. When it is properly cleaned, replace the battery. All these steps are to avoid delayed failure caused by corrosion. Best thing to do would be to clean it with an ultrasonic cleaner, but if you do not have access to one, this will at least help. Once all this is done, reassemble your phone and reevaluate. Until it is cleaned, everything will only be a guess. Even with all this, water damage has only a small chance to be fully repaired, but at least you can give it a try. From your description it is possible that this is just your display, but it could also be water damage on the logicboard that causes it.