  Motivated by applications to congested traffic problems, we establish higher
integrability results for the gradient of local weak solutions to the strongly
degenerate elliptic PDE $-\mathrm{div}\left((\vert\nabla
u\vert-1)_{+}^{q-1}\frac{\nabla u}{\vert\nabla
u\vert}\right)=f,\,\,\mathrm{in}\,\,\Omega$, where $\Omega$ is a bounded domain
in $\mathbb{R}^{n}$ for $n\geq2$, $1<q<\infty$ and $\left(\,\cdot\,\right)_{+}$
stands for the positive part. We assume that the datum $f$ belongs to a
suitable Sobolev or Besov space. The main novelty here is that we deal with the
case of subquadratic growth, i.e. $1<q<2$, which has so far been neglected. We
also infer the higher integrability of the solution to a variational problem,
which is characterized by the above equation. For the sake of completeness, we
finally give a Besov regularity result also in the case $q\geq2$.
