1) Beat in an endurance contest 2) Gaze 3) Look longer 4) Outface 5) Stare 6) Stare down 7) To outdo in staring 8) Triumph in a certain contest 9) Win a no blinking contest 10) Win an eye battle Found on https://www.crosswordclues.com/clue/outstare
Out·stare' transitive verb To excel or overcome in staring; to face down. « I would outstare the sternest eyes that look.» Shak. Found on http://www.encyclo.co.uk/webster/O/47