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 $$ -\,\textrm{div}\left( (\vert Du\vert -\lambda )_{+}^{p-1}\frac{Du}{\vert Du\vert }\right) =f\,\,\,\,\,\,\,\text {in}\,\,\Omega , $$ where $$1<p<\infty $$ , $$\Omega $$ is an open subset of $$\mathbb {R}^{n}$$ for $$n\ge 2$$ , $$\lambda $$ 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 $$B^{(p-2)/p}_{p',1,loc}(\Omega )$$ when $$p>2$$ , and that $$f\in L_{loc}^{{\frac{np}{n(p-1)+2-p}}}(\Omega )$$ if $$1<p\le 2$$ . 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 Clop et al. (Bull Math Sci 13(12):2350008, 2023) and Irving and Koch (Adv Nonlinear Anal 12(1):20230110, 2023).