I need to find the word Lämmönmyyntipalvelut from the database. Only, in the database it is in a field, whose value has been a PHP array, converted into JSO
The backslash is a meta character, MySQL understand it this way : "remove the next character and don't parse it as a meta-character".
So you need to escape the backslash :
SELECT * FROM table WHERE (services LIKE '%L\\u00e4%')
Now, MySQL will replace "\\" by "\" (the first backslash is a meta character used to escape the second one)