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

We consider local weak solutions to PDEs of the type (Formula presented.) where, is an open subset of for, is a positive constant and stands for the positive part. Equations of this form are widely degenerate for and widely singular for. 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 when, and that if. 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, 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).