 Solution

For the perpetual American call option, substitute a power $S^q$ into the stationary Black-Scholes equation with continuous stock dividends. The characteristic equation isIts two roots have negative product $-2r/{\sigma }^2$, so one is negative and one positive. The condition at zero excludes the negative power. Since $F(1)=-y<0$ and $F(q)\to \infty$ as $q\to \infty$, the positive root satisfies $p>1$; explicitly,Value matching gives $C(S)=(\hat{S}-K)(S/\hat{S}{)}^p$. The smooth fit condition gives $p(\hat{S}-K)/\hat{S}=1$, henceAs $y\downarrow 0$, the positive root tends to $1$ from above, so $\boxed{p\to 1,\hat{S}\to \infty }$. For fixed $S$, the value tends to $S$. In the dividend-free perpetual limit no finite threshold attains that supremum; this agrees with the finite-maturity no early exercise of a call without dividends.