Contest game: second order condition satisfied, but negative profits?
Well it says in the paper on p. 423:

"Thus the sufficiency assumption implies a unique pure strategy equilibrium. And if the sufficiency assumption is violated, at least one player will receive a negative payoff; that player can improve by reducing his or her effort to zero, which will destroy the equilibrium." [emphasis mine]

So that player will not make any effort because he/she cannot win. I leave it to you to figure out what the best reply from the other player is in that case. 

I guess to be completely mathematically correct one would have to formulate these kind of games with non-negativity constraints, but because it complicates the analysis they're often left out and an interior equilibrium is either assumed or, as here, checked afterwards.