My kenmore elite gas dryer model 110 75922401 ignites but doesn't stay on long enough to dry the clothes. Please advise what the problem may be and hopefully a solution. Thank you

Doesn't stay on long enough or the clothes aren't dry at the end of the cycle? Check the vent where it exits the house. Open? How is the air flow and when was the last time you cleaned the vent line out. I did one for a customer last week and pulled a half a bucket of wet lint out of a 15 ft. line. Lint screen clean? The most common drying issue is from a blocked vent line.

You will have to clean the drier out thoroughly. Every air duct will have to be removed and scraped and washed out to remove baked on lint deposits which is like concrete lining. This will burn so the manufacturer has installed the safeties to keep from having a house fire. Driers were the number one cause of fires at one time. Look at: http://www.youtube.com/watch?v=MMP7BW1lLs4 to see how. Also: http://www.youtube.com/watch?v=iBiRw-KElC4 ; http://www.youtube.com/watch?v=yFTZGgVisCI
When you finish cleaning the drier it is best to also replace the belt, too, because you do not want to have to disassemble it again, and if the belt is over two years old you should replace it.

Note that on most of the videos it does not show them washing out the parts, but this is important because it's the only way to really get the parts clean, just don't get motor wet or the capacitor if it has one. Vacuum out motor. I would move the drier out to your garage to have plenty of space to work. You also want to check your vent and clean it, also. The job is really easier than it sounds. There is no shortcut that will work; it has to be cleaned well. Before you re-assemble it make sure to use white lithium grease (high-temp grease) and grease all bearings. Just vacuuming out the dryer a little does little good. You also want to check the vent and clean it out. When you are done you will have a new dryer and it will dry like new again. You want to clean it like this every two years or every year with many children. DrTom4444