Do I have to expect snow at 3900 meters above sea level during summer?

I would definitely come prepared for snow at that elevation in late May.

As a possibly useful guide, you could look at the historical closing dates for Tioga Road. Tioga Road, the only east-west thoroughfare through the park, is closed by snow during the winter, and typically does not reopen until May or June. (Note that the reopening date is not the day when all the snow had melted; it's the day they were finally able to finish plowing it.) The high point of Tioga Road is Tioga Pass at 3031 meters, and you will be 900 meters higher than this.

This is also important if you were planning to travel on Tioga Road to get to your trailhead; it won't necessarily be open in time for your trip.

The exact answer will vary from year to year depending on the weather, both in terms of the volume of snowfall and how warm the weather is.

For the most part, the majority of snow in and around the Yosemite valley will have melted by the time you're talking about. The National Parks Yosemite website specifically states that :

The more moderate elevations, around 7,000 feet, typically melt by mid-May while the higher elevations, 10,000 feet and up, usually aren't accessible until mid-June or later.

Given that the majority of the park is below 7000 feet (Glacier Point is 7200 feet, El Capitan is 7500) that means that basically everywhere will be well clear of snow by the time you're there. Even Half-Dome at 8800 feet is generally clear of snow by the time they put up the 'cables' in mid/late May.

The best source of information about conditions is normally the NPS website linked above - and you can even check the conditions yourself for many of the locations on their Webcam page.