Best Response Functions for Cournot For a Cournot Duopoly game with Linear Demand functions, we need to find the optimal quantity of each firm such that profit is maximised. Given the linearity of the Demand curves, we will obtain a quadratic profit function. Since profit function is strictly concave and quasiconcave, we can ensure a unique maximum, i.e. a unique optimal quantity for the firm to produce. The above explaination is for a 'standard' game with Linear Demand function, say, $max(a-bq,0)$, with cost function, say, $C_i=c_iq_i$ for, say, a duopoly, $q=q_1+q_2$

What I need to understand is that if we have non-linear demand function, such that we cannot ensure a unique optimal quantity for profit maximisation (which would be the Best Response function), how would the Nash Equilibria or Equilibrium be obtained? That said, is it possible for a Cournot game, or for that matter, any game to have more than one Best Response? 

I'm still a bit rusty on the concepts, please bear with me. Is what I'm trying to understand, right conceptually? What am I missing?
In general a best response function returns a set of best responses. This can be seen in much simpler games than Cournot.

To give a degenerate example, if a player is always indifferent between their strategies, their best response function will always return the set of all their strategies.

When you have best response functions that give sets with more than one element, Nash equilibrium happens when each player's play is contained in the other player's best response set.