Before running the query you could set the case_sensitive_like pragma to ON.
PRAGMA case_sensitive_like = boolean;
The default behavior of the LIKE operator is to ignore case for ASCII
  characters. Hence, by default 'a' LIKE 'A' is true. The
  case_sensitive_like pragma installs a new application-defined LIKE
  function that is either case sensitive or insensitive depending on the
  value of the case_sensitive_like pragma. When case_sensitive_like is
  disabled, the default LIKE behavior is expressed. When
  case_sensitive_like is enabled, case becomes significant. So, for
  example, 'a' LIKE 'A' is false but 'a' LIKE 'a' is still true.
This pragma uses sqlite3_create_function() to overload the LIKE and
  GLOB functions, which may override previous implementations of LIKE
  and GLOB registered by the application. This pragma only changes the
  behavior of the SQL LIKE operator. It does not change the behavior of
  the sqlite3_strlike() C-language interface, which is always case
  insensitive.
Then just run the query as it is.  
PRAGMA case_sensitive_like=ON;
select username from users where username like 'aDam%';
Test it here on Sql Fiddle.