My neighbors are lying to landlord. What can I do?

I live on a third floor apartment with my two children and my downstairs neighbors are constantly complaining to the landlord. First off, they complain that we are loud, even when we are not home (but I constantly hear them slamming doors, having sex, and fighting. - Try explaining the sounds of sex to a 6 and 8 year old!) Because of a storm we had, I was cleaning my balcony off, which included a bucket that had some water in it. The water got stagnant because of the heat. I dumped it into three layers of plastic shopping bags and a garbage bag to take it out. About 1/4 cup spilled down through the cracks. No one was on the balcony and it did not hit anything they had down there. Two minutes later the landlady calls and says I poured hot coffee on their dog. I have had it with their lies. (They claimed a weekend that we weren't home we poured beer on him intentionally). The landlord will not let me out of the lease early and as a unemployed single mother I simply cannot afford to move. I have tried killing them with kindness. I have tried ignoring them. They are continually making up lies on an almost daily basis now. They also like to spy. I feel like I am being harassed. What if anything can I do?