Some Kenmore Elite dryers do have a thermistor and may not start (or will only run for a few seconds) if this component has failed. Without a model number, it is impossible for me to see specific technical information about your dryer. If your dryer has a thermistor, it is normally located on the blower housing. It can be tested using a volt/ohm meter. You would need to
unplug the dryer
and access the thermistor. To test it, pull one wire off and measure the resistance across the leads of the thermistor. Most thermistors for Kenmore Elite dryers will measure between 10K ohms (10,000 ohms) and 12K ohms at room temperature. If the thermistor is way out of this range, then it is likely bad and will need to be replaced.
You can view a parts list diagram to determine the location of the thermistor on your dryer on the
website. Use your full model number in the search field to access the parts diagram and listing for your particular dryer.
Before checking the thermistor, I recommend several other basic troubleshooting checks. Open the dryer door to see if the drum light turns on. This will verify that the dryer has electrical power. Push the door switch in with the door open to see if the light turns off, this will normally verify that the door switch is working properly. The door switch can prevent the dryer from running.
If you cannot verify that the outlet is working using the drum light, I recommend checking it directly. If you have a gas dryer, plug a small lamp or appliance into the outlet to verify that it is working. If you have an electric dryer, check the breakers for the dryer outlet to see if they are tripped. The only way to accurately check the electric dryer outlet would be to use a volt/ohm meter to measure voltage as shown in the diagram below. NOTE: You should only check this live outlet voltage yourself if you are completely confident in your technical ability to safely measure it.
If you find no problems with the outlet or the door switch, another common problem that would prevent an electric dryer from running would be the thermal fuse. This fuse is normally on the blower housing beside the thermistor. You can check it in the same manner as the thermistor (dryer unplugged and one wire removed). The thermal fuse should measure near zero ohms of resistance (closed). If you measure infinite resistance (open), then the fuse is blown and would need to be replaced. NOTE: If the thermal fuse is blown, check the exhaust air duct system for a clog or restriction that could have caused the thermal fuse to blow.
These tips may help you diagnose and fix your dryer problem. If you need more help, resubmit your question with additional details and the full model number of your Kenmore Elite dryer.