Ok... Don't panic here. I know it's frustrating, but you can get through it. (^_^)
First you are going to need to make some decisions. If you don't want to go through any sort of repair shop or service (and I don't blame you if you don't, especially since I do all my own service), then we know that it's not important to preserve any official seals (that will void the warranty... I'm assuming it's already out of warranty, or you'd have taken it back to wherever you got it or sent it to Sony).
Now, you need to see what (if any) actual damage has occurred. To do this you are going to need a solid, flat work surface, some device nearby that is grounded (like a water faucet or other pipe, or computer that's plugged in), a small screwdriver (and I do mean pretty small) and I recommend a soft (white) cloth, as it helps keep things from slipping or rolling around.
Ok, you have your work area set up, make sure it's well lit, and you have preferably got some sort of lint-free cloth, or paper wipes (glasses wipes that are specifically lint-free are good, don't try to use tissues, and they will leave paper lint everywhere) to make sure the inside of the screen lens (the part that's molded into the face of the PSP's case) and to VERY CAREFULLY clean the LCD itself (this is only necessary if it has gotten dust or other foreign matter on it, if not leave it alone. It's too easily damaged) with feather soft brush motions... We'll leave that for later.
Now. If you do some searching on YouTube you can find some impressive vids of people disassembling their PSP to add a video output attachment, and can use them to see how the PSP comes apart. You won't be going as far as them, as you don't have any reason to remove the LCD screen. Mostly you are going to be removing the face plate (which you can replace any time you want by shopping online) and examining the contacts and the rubber shoulder 'springs' that are under the shoulder button.
Flip the PSP face down, remove the battery cover and battery and set them aside. Inside the battery compartment there are 2 labels, one big one small that says to the effect of 'damage to this seal will void the warranty of this unit' or something like that (I don't feel like looking, sorry). Since we already don't care, you are going to use whatever method you prefer to remove it. A VERY small flat tip screwdriver or small knife can be slid under the edge and used to pry the label up. Set it aside. You now have access to the last screw you'll need to get to. Now, remove the screws that are at the bottoms of each of the holes on the back of the PSP, then the two inside the battery compartment that are the same size (as both each other and the others from in the holes) they will all be black. There is one additional screw you need to remove, it's on the edge of the psp, directly under the screen through the silver border, and it's silver in color.
Turn the unit over and you should now be able to remove the faceplate. Gently do so and set it aside. Then remove the offending shoulder button and the rubber 'boot' under it. Check for damage, see if it seems to flex ok, if so, look at the sensor film that's stuck to the shoulder area under where the 'boot' was. If it seems to need cleaning, do so. Once everything is clean and looks ok (assuming it does), reassemble it (checking the LCD and lens to see if they got dust or anything on them cleaning if necessary) and test it.
If it continues to have the same problem, you are going to have to order replacement parts, but now you know what they look like and how they'll need to be installed in the unit.
Best of luck to you, I hope this helps. If so, please rate this answer appropriately. Thanks.
First you are going to need to make some decisions. If you don't want to go through any sort of repair shop or service (and I don't blame you if you don't, especially since I do all my own service), then we know that it's not important to preserve any official seals (that will void the warranty... I'm assuming it's already out of warranty, or you'd have taken it back to wherever you got it or sent it to Sony).
Now, you need to see what (if any) actual damage has occurred. To do this you are going to need a solid, flat work surface, some device nearby that is grounded (like a water faucet or other pipe, or computer that's plugged in), a small screwdriver (and I do mean pretty small) and I recommend a soft (white) cloth, as it helps keep things from slipping or rolling around.
Ok, you have your work area set up, make sure it's well lit, and you have preferably got some sort of lint-free cloth, or paper wipes (glasses wipes that are specifically lint-free are good, don't try to use tissues, and they will leave paper lint everywhere) to make sure the inside of the screen lens (the part that's molded into the face of the PSP's case) and to VERY CAREFULLY clean the LCD itself (this is only necessary if it has gotten dust or other foreign matter on it, if not leave it alone. It's too easily damaged) with feather soft brush motions... We'll leave that for later.
Now. If you do some searching on YouTube you can find some impressive vids of people disassembling their PSP to add a video output attachment, and can use them to see how the PSP comes apart. You won't be going as far as them, as you don't have any reason to remove the LCD screen. Mostly you are going to be removing the face plate (which you can replace any time you want by shopping online) and examining the contacts and the rubber shoulder 'springs' that are under the shoulder button.
Flip the PSP face down, remove the battery cover and battery and set them aside. Inside the battery compartment there are 2 labels, one big one small that says to the effect of 'damage to this seal will void the warranty of this unit' or something like that (I don't feel like looking, sorry). Since we already don't care, you are going to use whatever method you prefer to remove it. A VERY small flat tip screwdriver or small knife can be slid under the edge and used to pry the label up. Set it aside. You now have access to the last screw you'll need to get to. Now, remove the screws that are at the bottoms of each of the holes on the back of the PSP, then the two inside the battery compartment that are the same size (as both each other and the others from in the holes) they will all be black. There is one additional screw you need to remove, it's on the edge of the psp, directly under the screen through the silver border, and it's silver in color.
Turn the unit over and you should now be able to remove the faceplate. Gently do so and set it aside. Then remove the offending shoulder button and the rubber 'boot' under it. Check for damage, see if it seems to flex ok, if so, look at the sensor film that's stuck to the shoulder area under where the 'boot' was. If it seems to need cleaning, do so. Once everything is clean and looks ok (assuming it does), reassemble it (checking the LCD and lens to see if they got dust or anything on them cleaning if necessary) and test it.
If it continues to have the same problem, you are going to have to order replacement parts, but now you know what they look like and how they'll need to be installed in the unit.
Best of luck to you, I hope this helps. If so, please rate this answer appropriately. Thanks.