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We do not distinguish between short-run and long-run profit maximisation in monopoly.
This is because a monopoly is protected by barriers to entry, which prevent new firms from entering the market to share in the abnormal profit made by the monopolist. Entry barriers enable the monopolist to preserve abnormal profits in the long run as well as in the short run.
By contrast, in perfect competition abnormal profits are temporary, being restricted to the short run.
The profit-maximising monopoly’s optimal output level, Q, is determined by where the MR and MC curves cross. The abnormal profit that the monopoly makes is shown by the shaded rectangle. Since there are barriers to the entry of new firms, these abnormal profits will not be competed away in the long run.