I checked our your site on mobile and I do see what you mean.
It does drop down but then snaps closed right away.
This is most likely due to a conflict with a 3rd party plugin you have installed.
The standard procedure to debug an issue like this is to disable all the plugins, except the ones required by the theme until the problem goes away. Then you will have identified the problematic plugin.
Please let me know when you find the plugin in question or if you need any further help.