  We give a new type of sufficient condition for the existence of measures with
maximal entropy for an interval map $f$, using some non-uniform hyperbolicity
to compensate for a lack of smoothness of $f$. More precisely, if the
topological entropy of a $C^1$ interval map is greater than the sum of the
local entropy and the entropy of the critical points, then there exists at
least one measure with maximal entropy. As a corollary, we obtain that any
$C^r$ interval map $f$ such that $h_{{\rm top}}(f)>2\log\|f'\|_{\infty}/r$
possesses measures with maximal entropy.
