Consider a monopoly with price power in the market and the demand is a function of price. Can the result of such a monopoly problem be called a nash equilibrium?
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In a somewhat degenerate way, yes. The specified demand function is a trivial case of a best response function describing the optimal quantity demanded for any given price of the monopolist. The monopolist's optimal profit-maximizing price in turn is the best response facing this demand. So you end up with a Nash equilibrium, where each player plays his/her best response and the choices are mutually consistent. Put differently, no player has any incentive to deviate.
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$\begingroup$ Relatedly, one can have also a competitive market where the monopolist price is a Nash equilibrium. See the Diamond paradox from industrial organization, where there are nonnegligible search costs to finding the best price. $\endgroup$ Mar 2, 2022 at 21:10
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$\begingroup$ Why is this »degenerate« ... whatever that means in an economic context? $\endgroup$ Mar 3, 2022 at 15:03
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$\begingroup$ Because I would not consider this your typical case of the intersection of two fully-fledged best response functions. $\endgroup$– jpfeiferMar 4, 2022 at 17:55