I previously wrote the lambda_web crate, but recent axum and lambda_http can be integrated directly.
This repository is a sample that uses axum and lambda-http. By separating the generation of axum::Router into a library crate, it can run on Lambda, a regular server, or integration tests. I recommend such implementation as it makes your code portable.
This sample is based on axum=0.7, hyper=1, lambda_http=0.9 combination. If you need older axum=0.6 sample, you can find it on axum06 branch.