Hi there, I just have a few comments that I hope can help you.
What is your new competition offering? Is it simply the same service, and they are undercutting you?
Is there anything you can offer that we would be the least challenged to meet? Your competitive advantage needs to be defensible, as well as convincing.
In my experience, the best ways to develop competitive edges is to make your service clearly better than the others. You could try offering additional services in addition to your core one of junk removal. Do you have any experience at other services that you could bundle? Marketing your service as the more organized, experienced and/or cheaper one, is hard to actually defend.
From what you said in your first post, it sounds like you were/are one of the first companies on the scene, so you hopefully you have an established group of loyal clientele. Try to encourage word-of-mouth advertising or referrals to take advantage of this.
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