Let’s cover a few basics:

1) Lydian is a major mode. It is determined major because it contains a Major 3rd. In the key of A Lydian, the major 3rd is C#. (If it was C instead of C#, then it would be some kind of minor mode. ie., phrygian, dorian, aeolian.)

2) Since it is a major, you can use A major pentatonic right along with it.

Example 1:
This lick combines sweep picking and tapping in conjunction with the Lydian mode.The first 5 notes of lick #1 are played with a single downward stroke. This is called sweep picking. The tapping is done with the third and middle finger. I find this use of multiple fingers works best. This enables me to use my palm to mute the string noise and my pick to pick any notes that are not tapped. Work on this technique slowly. It can be very difficult to get down. The chord in the background is an A Major. The notes that are picked in the background are D#, A, and E.

Example 2:
This lick takes the techniques used in all the last lick a little further. The notes that are not tapped are played by using sweep picking. If the notes ascend, they use a downward sweep. If the notes descend, they use an upward sweep. This covers this technique in more detail. Again, the tapping is played with the third and middle fingers. (3rd finger taps the D# note on the 23rd fret). Also the notes that are tapped on at the end (A-24th, G#-23rd) are natural harmonic’s. Although it is not notated, while you tap these notes on, you will hold the same note one octave below it on the same string (that would be the 11th and 12th fret). This lick is played against an A power chord (A, E). The effect that was used on this lick is a phaser.