  We study the limit behavior of differential equations with non-Lipschitz
coefficients that are perturbed by a small self-similar noise. It is proved
that the limiting process is equal to the maximal solution or minimal solution
with certain probabilities $p_+$ and $p_-=1-p_+$, respectively.
  We propose a space-time transformation that reduces the investigation of the
original problem to the study of the exact growth rate of a solution to a
certain SDE with self-similar noise. This problem is interesting in itself.
Moreover, the probabilities $p_+$ and $p_-$ coincide with probabilities that
the solution of the transformed equation converges to $+\infty$ or $-\infty$ as
$t\to\infty,$ respectively.
