Something is wrong with my laptop's battery. If it is under warranty (until next month), would they charge me?

I currently have a Lenovo yoga 2 pro. The first issue I had with the laptop was that whenever it would update, it would take HOURS to do so. The last time it updated, it was stuck on the screen, so I had to completely reset my laptop. Then everything is working.
Things were working fine for two months, until one day I realized that the laptop wasn't charging even though it was plugged in. So I went out and bought another AC adapter (which is EXPENSIVE), and it started working again for about 2 weeks. Recently it hasn't been working, so now I have to PUSH the AC adapter into my laptop and hold it there in order for it to charge… But now today, it's not charging at all.
I figured I might as well get it completely fixed since it's still under warranty. But where do I go? And would I have to pay?

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