Hello! I'm trying to figure out how to find the best response functions and a Nash equilibrium when given a utility (profit) function that has a quadratic in the denominator.

For example: Consider the following version of a Cournot duopoly. Each firm produces $\displaystyle q_i$ ≥ 0. There is a constant production cost, $\displaystyle c$. The resulting profit (utility) functions are:

$\displaystyle u_1(q_1,q_2)=q_1/(1+(q_1+q_2)^2) - c$

$\displaystyle u_2(q_1,q_2)=q_2/(1+(q_1+q_2)^2) - c$

1) Find the best response function for each firm.

2) Find a Nash equilibrium or show that there is none.

Thanks!