An enhanced auto-ignition in a gasoline internal combustion engine, comprises a fuel injector directly communicating with said combustion chamber for spraying gasoline fuel. The fuel injector sprays a first injection quantity of gasoline fuel into a combustion chamber at first fuel injection timing, which falls in a range from the intake stroke to the first half of the compression stroke, thereby to form air / fuel mixture cloud that becomes a body of mixture as the engine piston moves from the first fuel injection timing toward a top dead center position of the compression stroke, and the fuel injector sprays a second injection quantity of gasoline fuel into the body of mixture at second fuel injection timing, which falls in the second half of the compression stroke, forming mixture cloud that is superimposed on a portion of said body of mixture, thereby to establish the cylinder content wherein the density of fuel particles within the superimposed portion is high enough to burn by auto-ignition at an ignition point in the neighborhood of the piston top dead center position of the compression stroke, causing temperature rise and pressure, which initiate auto-ignition of the fuel particles within the remaining portion of said body of mixture.