My Leapster 2 turns off after "cartridge game" is

SOURCE: Leapfrog Leapster touch screen

Press the ? button and hold it while turning unit on.
Press the "a" button to select touch calibration
Press each + with the stylus
Restart machine

SOURCE: After charging our green Leapster 2. We turned it

You can try calibrating the stylus pen before trying to replace it. Hope this works :) Here are step by step instructions straight from the Leapster2 website:

If your Leapster2 does not respond to the stylus pen, it needs to be calibrated. Calibrating the Leapster2 is easy:

1) Ensure that a game cartridge is not inserted in the Leapster2

2) Hold down the Hint (?) button while turning the unit on. Continue to hold the Hint button down until the LEAPSTER2 menu appears. Remember that you need to hold the "Hint" button down the whole time while you are pressing the power button, and releasing it.

3) The LEAPSTER2 menu will show the options Base Info, Cart1 Info, Cart2 Info, Cart3 Info, and Touch Calib. If you do not see this menu after powering on the Leapster2, turn the Leapster2 off and repeat steps 1 & 2 until you see the menu.

4) When the LEAPSTER2 menu screen appears, press the A button on your handheld.

5) Using the stylus pen, touch each of the crosshairs that appear

6) Restart unit

SOURCE: I turn on my daughter's

You have to make sure that your Cartridges is clean. Try to vacum of blow both of the cartridge hole and also the place where you put the cartridge, Sometime very small tiny dirt will makes that the cartridge cannot be recognise by the console.

Instead of 1 starter game and a few demo game that you get from Leapfrog Connect, you can also get more downloadable games for your Leapster 2. You can get it here:
leapster-2.blogspot.com