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my brother's PSP 2004 when switched on makes noises as if it were working normally but the screen remains dark. Pushing the screen light button I can see some light changes in illumination.

Does somebody know about this problem?
If I were sure, I would order a new screen ...
Is there a known method of testing the display unit?

I don't think there is a safe method of resolving this question without a PSP LCD you know works. Did anything happen that made the PSP screen fail? Did it drop? Did it get wet? Maybe you can deduct from there if it is more likely to be the logic board or the LCD. I would recommend opening it up. If you don't see any physical damage to the logic board, chances are around 80% the issue is caused by the screen.