I do not want to overpost to try to steel this question, so I will send this as a request so He (JR) can get paid.
DO NOT JUMPSTART ANY MOWER OR DAMAGE WILL OCCURR TO THE CHARGING SYSTEM! You most likley have a bad battery. It probably has a bad cell. A battery with a bad cell will not jump-start correctly anyway. Have the battery checked. and make sure it has at least 340 CCA. Do not use a discount store battery. They many times are bad when you get them. You also could have a bad ground wire. Clean the battery cables at both ends. Most solenoids that are replaced do not need to be replaced.
Small Engine Technician
Owner of outdoor power equipment business, repairing all makes and models.