Hi!
If close the keyboard the input box loses focus and so the ajax results disappear
That should not happen at all. I have tried a few mobile devices, including iPhone, iPad and android 6, 7 phones but the input field does not blur. I suspect this is either a device behavior thing, or more likely a script that somehow detects a screen size change or virtual keyboard behavior in some way, and blurs focus from input fields.
Can you please try the demo site, and let me know if you see that behavior there as well: https://wp-dreams.com/demo/ajax-search-lite/
If so, then please let me know what device (and browser) are you using for testing, I will try to emulate it somehow, if I can’t get a hold of it.
In earlier versions of the plugin there was a solution that would try to hide the virtual keyboard whenever the results list appeared, however apple decided to remove the control of the virtual keyboard event support, and the android market sort of adapted to it. Still, at this point there is no working cross-device solution to this issue.
Instead, search engines no longer interact with the keyboard state, and let the user close it as it fits.
However it should still not lose focus on closing the keyboard, that is not intended.