Value of an American put that never expires

A non-dividend stock follows geometric Brownian motion with volatility $\sigma$, and the risk-free rate is $r > 0$. Find the value of an American put with strike $K$ and no expiry date.

$P(S) = (K - S^*)\left(\frac{S}{S^*}\right)^{-\gamma}$ for $S \ge S^*$ (and $K - S$ below it), with $\gamma = 2r/\sigma^2$ and $S^* = \frac{\gamma}{1+\gamma}K$. With no time dependence the Black-Scholes PDE becomes an ODE, $\tfrac12\sigma^2S^2P'' + rSP' - rP = 0$, whose bounded solution is $AS^{-\gamma}$. Value matching and smooth pasting at the exercise boundary $S^*$ pin down $A$ and $S^*$.