MySQL search function join on multiple conditions
At the moment you're searching for e.g. "Barney Likert" (the whole string) in both the studies and the elements columns you've specified. 

You need to either have two search strings, one for searching studies, and one for searching elements. Or if your users can't be bothered to use two separate fields, then you could parse the string before it's sent to the database, split it up if it consists of two words, and execute a particular query depending on what the search string is. E.g. you could have one query for input of one word, and another query for input of two words. 

You may also want to consider changing the JOIN to LEFT OUTER JOIN, otherwise you will only get results from studies that have at least one element. If you know that all studies have at least one element, then you can keep it as is, as this type of join is faster.