I have a multi entry visa issued by Denmark Embassy which is valid from 1st May to 23rd May. It has max number of days as 8. I would be staying in Denmark for 6 days only from 1st May to 6th May.
Now, I want to visit Austria from 16th May to 18th May and Germany from 31st May to 2nd June. Can I apply for another Visa now? I am worried that my Denmark Visa and new Visa will have overlapping dates.
Since your current visa is not sufficient for your planned further visits (you'll have one day too few for the 3-day Austria visit), you'll need to apply for a new visa. This is standard procedure.
One sometimes sees claims that it is impossible or "illegal" to have two overlapping Schengen visas in one person's passport. However, when this has come up in previous questions we have been unable to find any legal enactments to that effect. Consulates will not issue visas to people who don't need them due to already having a visa that allows their intended trip, but that is not your situation.
Apply to Austria with full details of your plans, pointing out that due to your visit to Denmark your existing visa will not suffice for the trip to Austria. If your application is approved, the consulate will either revoke your Danish visa and issue a new one covering all three trips, or issue an overlapping visa. You don't need to choose which of these you want when applying; it is up to the consulate to determine how your plans will fit in their systems.
