Assuming the firms are price takers, the profit function is Profit(q) = (p-a-bq)q, the first order condition (FOC) is p-a = 2bq and the second order condition (SOC) is b > 0. Substituting the FOC into the profit function gives Profit(q) = b.q^2 so profit is negative unless the SOC is satisfied. The representative firm charges
p=(J.a.beta+2.alpha.b)/(J.beta+2b) and sells q=(alpha-a)/(J.beta+2b). Market supply is J.q*. Since Profit(q)>0 if SOC is satisfied, J goes to infinity, p to alpha and q to zero.
Assuming the firms are Cournot oligopolists, the firm charges p=alpha-beta(alpha-a)/2(b+beta.J) and sells (alpha-a)/2(b+beta.J) with profit function (alpha-a)^2/4(b+beta.J)>0 as the SOC is b+beta.J>0 so b can be negative in this case. J, p and q as before.