1.e4 c5 2.Nf3 d6 3.Bc4 Nc6 4.d4 e×d4 5.c3 does this gambit exist? If no, can someone help me find a name for it pls?
Invent your own opening!!

1.e4 c5 2.Nf3 d6 3.Bc4 Nc6 4.d4 e×d4 5.c3 does this gambit exist? If no, can someone help me find a name for it pls?
It exists, and has no special name.

Italian Game: Romanian Gambit, Cleveland Countergambit
Italian Game: Romanian Gambit, German Variation
The Quagmire Gambit, it's a variation of the "Queens Gambit Accepted, Central Variation, Alekhine System' that gambits the e4 pawn with knight f3.
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This is the definition of hope chess: Even a dummy would easily see that 5...Bg4? is an elementary blunder because of 6.Bxf7+ (why should 6.Ne5 be better? Black's position is surely enough a mess after the forced 6...Be6, but white isn't winning outright, as is the case after 6.Bxf7+).
If Black does not fall for this cheapo and play a normal move like 5...e6 or 5...g6, white will be struggling to find compensation for the pawn.