How to catch the error when inserting a MongoDB document which violates an unique index?

You will need to test the error returned from the save method to see if it was thrown for a duplicative username.

app.post('/authenticate', function(req, res) {
  var user = new User({
    username: req.body.username
  });

  user.save(function(err) {
    if (err) {
      if (err.name === 'MongoError' && err.code === 11000) {
        // Duplicate username
        return res.status(422).send({ succes: false, message: 'User already exist!' });
      }

      // Some other error
      return res.status(422).send(err);
    }

    res.json({
      success: true
    });

  });
})

You can also try out this nice package mongoose-unique-validator which makes error handling much easier, since you will get a Mongoose validation error when you attempt to violate a unique constraint, rather than an E11000 error from MongoDB:

var mongoose = require('mongoose');
var uniqueValidator = require('mongoose-unique-validator');

// Define your schema as normal.
var userSchema = mongoose.Schema({
    username: { type: String, required: true, unique: true }
});

// You can pass through a custom error message as part of the optional options argument:
userSchema.plugin(uniqueValidator, { message: '{PATH} already exists!' });

Try this:

app.post('/authenticate', function(req, res) {
        var user = new User({
            username: req.body.username
        });

        user.save(function(err) {
            if (err) {
                // you could avoid http status if you want. I put error 500 
                return res.status(500).send({
                    success: false,
                    message: 'User already exist!'
                });
            }

            res.json({
                success: true
            });

        });
    })