JAVA_HOME not set in script when run using sudo

For security reasons, sudo may clear environment variables which is why it is probably not picking up $JAVA_HOME. Look in your /etc/sudoers file for env_reset.

From man sudoers:

env_reset   If set, sudo will reset the environment to only contain the following variables: HOME, LOGNAME, PATH, SHELL, TERM, and USER (in addi-
           tion to the SUDO_* variables).  Of these, only TERM is copied unaltered from the old environment.  The other variables are set to
           default values (possibly modified by the value of the set_logname option).  If sudo was compiled with the SECURE_PATH option, its value
           will be used for the PATH environment variable.  Other variables may be preserved with the env_keep option.

env_keep    Environment variables to be preserved in the user's environment when the env_reset option is in effect.  This allows fine-grained con-
           trol over the environment sudo-spawned processes will receive.  The argument may be a double-quoted, space-separated list or a single
           value without double-quotes.  The list can be replaced, added to, deleted from, or disabled by using the =, +=, -=, and ! operators
           respectively.  This list has no default members.

So, if you want it to keep JAVA_HOME, add it to env_keep:

Defaults   env_keep += "JAVA_HOME"

Alternatively, set JAVA_HOME in root's ~/.bash_profile.

Run sudo with the -E (preserve environment) option (see the man file), or put JAVA_HOME in the install.sh script.