I will not go into details on how to get a PhD scholarship in US. All I did in my time was look universities up and go through their application process.
Related to what you're saying, it could go into your statement of research.
I would be skeptical to anyone trying to contact me and tell me they solved some important question in a field. Professors in US, and even postdocs get that kind of letters every day. Most of them are from mentally deranged people who simply have no scientific background. But if you're not one of those people, the chances that a potential supervisor would ignore your mail as scientific spam are high, precisely because of those people.
But, I remember I taught some course at a summer school this year. The way it was organized, I got to select the applicants I would work with myself. After I carefully read all the normal things (CVs, letters of recommendation, etc.) I made my final decision based on their research interests as they were reflected in their statements.
They were young people and they were gushing about how much they love their field of study and what interests them and so on, but I only looked at how well they explained what they did themselves and how. And I think I selected the best applicants.
So, if you think you solved an important problem, you should talk about it in your research statement. You should be succinct and honest. Even if your solution is not as great or as important as you originally thought, if you present your main result in a clear and logical manner, with easy to follow arguments, your application will be very good. If you already are in contact with some strong researchers, ask them for a model of a cover letter they sent when they published their work in a top journal. That's how you should present your research result in your statement.
Another thing that comes to mind is that you could have already submitted your research to a top journal, if it is as good as you say. Even if it isn't it's worth getting the opinion of an editor and of some referees. A good published work, at this stage in your career, would help your application a lot.
Finally, if you wish to contact a potential adviser, you could sent them an email saying you applied to their school's graduate program, and given your research interest, and their expertise in your field, you think you would be a good fit in their research group. You tell them what you've been working on, and attach that research statement, if they want to read it. If they are looking to hire a PhD student, they might be happy to have you, especially if your work is as high quality as you think.