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

Abstract

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 $1 2$, 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].