Prove the uniqueness of steady state
Rearranging the steady state equation
$$
\overline{p}^{\alpha}=\alpha y\overline{p}^{\alpha-1}-
\alpha\overline{p}^{\alpha}-\frac{a+1}{\sigma}
$$
we get
$$
(1 + \alpha)\overline{p}^{\alpha}=\alpha y\overline{p}^{\alpha-1}-
\frac{a+1}{\sigma}.
$$
As $\alpha \in [0,1]$, the left hand side of the equation is increasing in $\overline{p}$ and the right hand side is decreasing. At least one of these is strictly monotonic because $\alpha$ cannot be $0$ and $1$ at the same time. Hence at most one solution is possible.