Using a standard PDE approach to price an American perpetual put option I obtain that the price of such option has the following form: $$ V(S) = A S + B S^{-2r/\sigma^2}. $$ And then I need to find a proper $A$ and $B$ coefficients to have the final solution. Finally I receive: $$ V(S) = \frac{K\sigma^2}{2r + \sigma^2}\left(\frac{S}{K} \frac{2r + \sigma^2}{2r}\right)^{-2r/\sigma^2}, \quad S \geq S^{*} = \frac{K}{1+\frac{\sigma^2}{2r}}. $$
This result is taken f.e. from 'Paul Wilmott on Quantitative Finance' book.
My question is:
Why I can not use the same technique to price American perpetual call option? When I apply the same method I obtain that my price has a form: $$ V(S) = A S. $$ But I am not able to derive that the coefficient $A$ should be equal to $1$.
Can anybody explain me where is the key issue of this problem?