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

We consider local weak solutions to PDEs of the type − div ( (|Du| − λ) p−1 + Du |Du| ) = f in Ω, where 1 < ∞, Ω is an open subset of Rn for n ≥ 2, λ is a positive constant and (·)+ stands for the positive part. Equations of this form are widely degenerate for p ≥ 2 and widely singular for 1 < 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 (Ω) when p > 2, and that f ∈ L np n(p−1)+2−p loc (Ω) if 1 < p ≤ 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 λ = 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).