= Solution
For $S>b$, maximize the coefficient $H(b)=b^\gamma(K-b)^n$ of $S^{-\gamma}$. Its logarithmic <derivative> is
$$
\frac{H'(b)}{H(b)}=\frac\gamma b-\frac n{K-b}.
$$
It changes sign once, at $b=\gamma K/(\gamma+n)$. The <perpetual power put option> therefore has candidate value
$$
\boxed{b=\frac{2rK}{2r+n\sigma^2},\qquad V_n(S)=\begin{cases}(K-S)^n,&0\leq S\leq b,\\(K-b)^n(S/b)^{-2r/\sigma^2},&S>b.\end{cases}}
$$
The same boundary follows from <smooth fit>, $-\gamma(K-b)^n/b=-n(K-b)^{n-1}$. To establish optimality among all <stopping times>, rather than only fixed barriers, we verify the candidate. Let $\mathcal L=rS\partial_S+\sigma^2S^2\partial_{SS}/2$ be the <infinitesimal generator>. The maximum property of $H$ gives $V_n\geq g_n$ everywhere, and $(\mathcal L-r)V_n=0$ for $S>b$. For $S\leq b$, set $y=S/(K-S)\leq\gamma/n$. Then
$$
\frac{(\mathcal L-r)g_n}{g_n}=\frac12\sigma^2\big[n(n-1)y^2-\gamma ny-\gamma\big]\leq-r(1+y)<0.
$$
Thus the discounted candidate is a nonnegative <supermartingale>. The <smooth fit> removes the derivative-jump <local time of a semimartingale> term in the <Itô formula>. The <optional stopping theorem> and localization bound every exercise policy's discounted <expected value> by $V_n$. Immediate exercise attains it below $b$; the first hitting time of $b$ attains it above $b$. The terminal remainder vanishes since $0\leq e^{-ru}V_n(S_u)\leq K^n e^{-ru}\to0$. This proves the displayed value.
For $n=1$ it reduces to the ordinary <perpetual put option>. Larger $n$ lowers the optimal trigger, because the more strongly curved <financial payoff> makes further downward moves more valuable. If $r=0$, under the <risk-neutral measure> the stock tends to zero <almost surely>, so the supremum is $K^n$, approached as the exercise time tends to infinity and not attained at a finite time when $S>0$. If $r<0$, the values at deterministic exercise times grow without bound; the perpetual value is infinite. These cases do not admit the positive-rate formula above.
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