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

We consider local weak solutions to PDEs of the type $$-\mathrm{d}\mathrm{i}\mathrm{v}((\mathrm{\mid }Du\mathrm{\mid }-\lambda {)}_{+}^{p-1}\frac{Du}{\mathrm{\mid }Du\mathrm{\mid }})=f\text{in}\mathrm{\Omega },$$ where $1<p<\mathrm{\infty }$, $\mathrm{\Omega }$ is an open subset of ${\mathbb{R}}^n$ for $n\ge 2$, $\lambda$ 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 $B_{p^{\mathrm{\prime }},1,loc}^{(p-2)\mathrm{/}p}(\mathrm{\Omega })$ when $p>2$, and that $f\in L_{loc}^{\frac{np}{n(p-1)+2-p}}(\mathrm{\Omega })$ if $1<p\le 2$. 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].