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Ian- The short answer is no. The elevator will need the same amount of energy to climb the cable regardless where that energy comes from. Solar panels on the elevator itself are the best way to get that energy. The two most obvious alternatives to solar panels on the climber are fueling the climber (gasoline, hydrogen, ect), or transmitting the power from the ground (which could be generated any number of ways. In both cases, the biggest problem is in the weight. The amount of fuel the climber would require would be significant, I am guessing somewhere in the tens of thousands of gallons. In the case of transmitting power, you would need a conductive cable. That would be a metal cable. In order to transmit the power needed to move the elevator, that cable would need to be fairly thick. Even having a metal cable as thin as speaker wire would snap under its own weight at the length required for the elevator. So no, it is not possible to send power up the cable.

a3rdleg, dinosaurs were around from approximately 250 million years ago until about 65 million years ago. this was known as the mesozoic era. there were 9 other eras in earths history, extending back to the formation of the earth around 5 billion or so years ago. life began in the second of these eras, at about 3.8 billions years ago. oil has been forming ever since. most of our oil today is pre-dino oil. most of it would probably have formed during the carboniferous period, lasting from 359 million to 299 million years ago. there would have been plenty of underground oil at the time of the mass extinction event that killed the dinosaurs.