Rough differential equations with power type nonlinearities

In this note we consider differential equations driven by a signal $x$ which is $gamma$-Holder 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 $kappain(0,1)$. Two different methods are used in order to construct solutions: (i) In a 1-d setting, we resort to a rough version of Lampertis transform. (ii) For multidimensional situations, we quantify some improved regularity estimates when the solution approaches the origin.