On the second-order regularity of solutions to widely singular or degenerate elliptic equations

We consider local weak solutions to PDEs of the type $$-\,\mathrm{div}\left((\vert Du\vert-\lambda)_{+}^{p-1}\frac{Du}{\vert Du\vert}\right)=f\,\,\,\,\,\,\,\text{in}\,\,\Omega,$$ where $1<p<\infty$, $\Omega$ is an open subset of $\mathbb{R}^{n}$ for $n\geq2$, $\lambda$ is a positive constant and $(\,\cdot\,)_{+}$ stands for the positive part. Equations of this form are widely degenerate for $p\ge 2$ and widely singular for $1<p<2$. We establish higher differentiability results for a suitable nonlinear function of the gradient $Du$ of the local weak solutions, assuming that $f$ belongs to the local Besov space $B^{(p-2)/p}_{p',1,loc}(\Omega)$ when $p>2$, and that $f\in L_{loc}^{{\frac{np}{n(p-1)+2-p}}}(\Omega)$ if $1<p\leq2$. The conditions on the datum $f$ are essentially sharp. As a consequence, we obtain the local higher integrability of $Du$ under the same minimal assumptions on $f$. For $\lambda=0$, our results give back those contained in [12,28].