I've had excellent results from cold-pressed Neem Oil (from Dyna-Gro, but any PURE cold-pressed Neem Oil will work). I mix 2 Tbsp. Neem and 2 tsp. Soap (not detergent, real soap) into 1 Gallon of water, and mist the plant, making sure to cover everything. This prevents sucking insects from reproducing, but doesn't hurt helpful insects. It has a faint garlic or onion odor, but it's not very offensive. Apply it twice a week for the first couple of weeks, and once every week or two after that. It will not immediately kill the aphids, it will take a couple of weeks for it to work, and you will need to continue to treat them. But it is safer than many alternatives. This will help with spider mites, too, which frequently show up when you're growing inside. As others have said, continue to try to spray them off the plant and squish them whenever you can. It will be an ongoing battle, an you will probably never get rid of them completely, but a couple here and there won't hurt your peppers. Good luck, and enjoy your peppers! :-)

I would have to say it is a bad case of sunburn. YOu probably put them outside for the first time and put them in direct sunlight. You have to harden them off gradually when going from indoors and artificial light to outside in direct sunlight. Start with putting them outside in the shade for a couple days for an hour of two per day. Then increase the sun exposure and the time outside gradually day by day. Your plant won't die, but some of the leaves will likely die off. No big deal. They will typically be the older leaves. So, at this point, you plants are probably hardened off pretty well even though you sun burned them in the process. It probably wont need any more hardening off. In the future, do it slowly and gradually. It should take a good week straight of gradually increasing their sun exposure. Also be careful of wind as the plants are not used to it and therefore they haven't yet built up a lot of strength in their stems. Wind exposure should be limited early on as well or you may find some toppled plants when you go to bring them back in. Bruce

Roots can stand a lot of jostling. Get a trowel in the pot and loosen the root ball around the edges. If it's not too root bound, you should be able to tease it out. Not critical yet. Hack that towering stem off when you can. It's not critical to do it right now, but it will make re-potting a lot easier. I would treat the spider mites first. You can pull off the webs with a paper towel then apply insecticidal soap. Read the directions and soak the little buggers. Your watering scheme is working - the plant has gotten this far - so keep at it for now. Try to get it more light. Closer to window and lower so all the leaves get some light. Peppers are tough. It will pull through. Dennis

This is from a link above. It reads well. If nothing else it is worth a notation, in my mind at least. Peppers Are Wilting: What To Do For Wilted Pepper Plants - If your peppers are wilting suddenly, developing large yellow areas and drooping (especially if this starts on the bottom leaves and moves upward) despite adequate watering, fungal wilt is probably to blame. Spotted wilt virus is a less common cause of wilted pepper plants, but if your plant’s leaves are dotted with brown or black spots or unusual yellow lines or circles and the symptoms move through the plant from the top down, it is very likely the cause. Occasionally, bacterial pepper wilt may affect your plants. Pepper plants will wilt and die quickly and upon inspection, the inner stems may be dark, watery and hollow.