To begin with, move static resources (images, css, js) to a server in Australia and keep the rest of your application in US. Since the majority of your webpage is going to be static resources, this is going to give you a significant improvement in response time.
If you can afford it, use a Content Delivery Network that has a presence in Australia. If you cannot afford one, use a regular hosting provider to host your files in Australia, and then switch between the two data centers using Any cast DNS. An ever cheaper solution is to determine the location based on ip address, and then use either http://us.mydomain.tld/image.jpg or http://au.mydomain.tld/image.jpg -- but this is obviously slower than the anycast DNS.
If the application is still slower, you will have to create a replica of your entire application in Australia, and that becomes a lot more expensive because you have to start replicating databases and deal with data synchronization.