The title of your thread is a little confusing. Perhaps a translation problem.
Phex cannot run without java.
Sorry I can't help you because I don't have linux mint (though perhaps Ubuntu is not dramatically different.) I have heard Phex will run with the open source java, so you can try installing that and making sure it's set as the priority java.
How-to Install Java OpenJDK JRE-JDK on Linux Mint 18.x 32-64bit
Alternatively see
Howto install Oracle (Sun) Java JRE for Linux Mint How to install Oracle JDK on Linux Mint though these refer to Oracle Java and also concentrate upon Firefox. Keeping in mind JDK is the developer version and is much larger.
Does that help? Or is Phex unable to find Java after java has been installed?