This is an assignment question. Not looking for a solution, but would appreciate a nudge.
Consider a 2-player non-zero sum game, where the strategy sets are U1 = U2 = [0,1] and let J1(u1,u2)=(u1)^3 - 3u1u2 - (u2)^3 [where J2 = -J1]. Determine the pure strategies that are Nash-Equilibrium. (Note that J is strictly convex in u1 and that for every u1(element of)U1, the max value of J with respect to u2 is attained either at u2=0 or u1=1