Look at the capacitors on the motherboard. They should all have flat tops, and the tops on the larger ones will probably have an X or a K etched into the tops. If any of the capacitors have bulging at the top, then you will either need a new motherboard or you might be able to get somebody to solder in new capacitors to replace the bad ones. If the capacitors leaked, you might see some gunk on the motherboard. In that case, you probably will not get much mileage from soldering in new capacitors.
Here's some pictures to show you what kind of thing you are looking for.
Leaking from bottom
These capacitors look normal; the tops are flat and not punctured. However they have leaked from the underside, spilling fluid onto the motherboard. In this case, the capacitors are probably not replaceable since the fluid could have damaged the motherboard.
The tops of these capacitors are slightly domed, meaning excess pressure has built up inside the capacitor. These capacitors may or may not still work properly, and they can be hard to notice. If there are any of these, it is possible they are causing the problem. It is likely possible to solder in a new capacitor.
The pressure in these capacitors built up until the safety vent popped. These may be hard to notice depending on how widely the vent has opened. This is the same problem as with bulged tops, only more pronounced. These capacitors are definitely dead. It may be possible to solder in a new capacitor.
The casing of this capacitor in the center has come off completely due to excess pressure. This kind of failure is very rare because of the Ks and Xs scored into the tops of the capacitors to provide a point for venting. This is definitely the easiest to notice because there will be two large coils of metal standing out. Like the leaking capacitors in the first picture, there might be too much damage to simply replace the capacitor.
Images from Wikipedia article Capacitor plague