m/^(11+)(\1)+$/
The regular expression:

(?-imsx:^(11+)(\1)+$)

matches as follows:
  
NODE                     EXPLANATION
----------------------------------------------------------------------
(?-imsx:                 group, but do not capture (case-sensitive)
                         (with ^ and $ matching normally) (with . not
                         matching \n) (matching whitespace and #
                         normally):
----------------------------------------------------------------------
  ^                        the beginning of the string
----------------------------------------------------------------------
  (                        group and capture to \1:
----------------------------------------------------------------------
    1                        '1'
----------------------------------------------------------------------
    1+                       '1' (1 or more times (matching the most
                             amount possible))
----------------------------------------------------------------------
  )                        end of \1
----------------------------------------------------------------------
  (                        group and capture to \2 (1 or more times
                           (matching the most amount possible)):
----------------------------------------------------------------------
    \1                       what was matched by capture \1
----------------------------------------------------------------------
  )+                       end of \2 (NOTE: because you're using a
                           quantifier on this capture, only the LAST
                           repetition of the captured pattern will be
                           stored in \2)
----------------------------------------------------------------------
  $                        before an optional \n, and the end of the
                           string
----------------------------------------------------------------------
)                        end of grouping
----------------------------------------------------------------------

m/^(11+){2,}$/
The regular expression:

(?-imsx:^(11+){2,}$)

matches as follows:
  
NODE                     EXPLANATION
----------------------------------------------------------------------
(?-imsx:                 group, but do not capture (case-sensitive)
                         (with ^ and $ matching normally) (with . not
                         matching \n) (matching whitespace and #
                         normally):
----------------------------------------------------------------------
  ^                        the beginning of the string
----------------------------------------------------------------------
  (                        group and capture to \1 (at least 2 times
                           (matching the most amount possible)):
----------------------------------------------------------------------
    1                        '1'
----------------------------------------------------------------------
    1+                       '1' (1 or more times (matching the most
                             amount possible))
----------------------------------------------------------------------
  ){2,}                    end of \1 (NOTE: because you're using a
                           quantifier on this capture, only the LAST
                           repetition of the captured pattern will be
                           stored in \1)
----------------------------------------------------------------------
  $                        before an optional \n, and the end of the
                           string
----------------------------------------------------------------------
)                        end of grouping
----------------------------------------------------------------------