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We study existence and uniqueness of solutions to the equation $dX_t=b(X_t)dt+ dB_t$, where $b$ is a distribution in some Besov space and $B$ is afractional Brownian motion with Hurst parameter $H\leqslant 1/2$. First, theequation is understood as a nonlinear Young equation. This involves a nonlinearYoung integral constructed in the space of functions with finite $p$-variation,which is well suited when $b$ is a measure. Depending on $H$, a condition onthe Besov regularity of $b$ is given so that solutions to the equation exist.The construction is deterministic, and $B$ can be replaced by a deterministicpath $w$ with a sufficiently smooth local time. Using this construction weprove the existence of weak solutions (in the probabilistic sense).  We also prove that solutions coincide with limits of strong solutionsobtained by regularisation of $b$. This is used to establish pathwiseuniqueness and existence of a strong solution. In particular when $b$ is afinite measure, weak solutions exist for $H<\sqrt{2}-1$, while pathwiseuniqueness and strong existence hold when $H\leqslant 1/4$. The proofs involvefine properties of the local time of the fractional Brownian motion, as well asnew regularising properties of this process which are established using thestochastic sewing Lemma.

Regularisation by fractional noise for one-dimensional differential equations with distributional drift