There's no legal method by which Trump could stay in office or extend his powers past January 20.
Trump can't extend his term directly: the Constitution as amended by the Twentieth Amendment states that the president's term ends on January 20, and there is no mechanism for changing that short of another amendment, a procedure that the President has no part in. The decision on who the next President will be is made by a joint session of Congress voting to accept the votes of the Electoral College, a procedure that, again, the President has no part in.
Executive orders are instructions to the executive branch on how to carry out laws passed by Congress. They can be freely revoked by the next President, they can be overturned by the courts, or they can be overruled or made irrelevant by Congress changing the law. Because of this, they can't be considered a reliable method of asserting power after the end of a President's term.