  In this note we consider differential equations driven by a signal $x$ which
is $\gamma$-H\"older with $\gamma>1/3$, and is assumed to possess a lift as a
rough path. Our main point is to obtain existence of solutions when the
coefficients of the equation behave like power functions of the form
$|\xi|^{\kappa}$ with $\kappa\in(0,1)$. Two different methods are used in order
to construct solutions: (i) In a 1-d setting, we resort to a rough version of
Lamperti's transform. (ii) For multidimensional situations, we quantify some
improved regularity estimates when the solution approaches the origin.
