Second-order regularity for weighted widely degenerate problems with explicit $u$-dependence

Abstract

We consider local weak solutions of widely degenerate elliptic PDEs of the type \begin{equation} \label{equazione mia} \mathrm{div}\Biggl(|x|^\beta(|Du|-1)^{p-1}_+\frac{Du}{|Du|}\Biggr)=\frac{|u|^{q-2}u}{|x|^\alpha} \ \ \text{ in }\Omega, \end{equation} where $2\leq p0$ are fixed exponents, $\Omega$ is an open subset of $\mathbb{R}^n,$ that contains the origin, $n>2,$ and $( \ \cdot \ )_+$ stands for the positive part. We establish a higher differentiability result for the composition of the gradient with a suitable function that vanishes in the unit ball for the gradient, under appropriate assumptions on the datum. The novelty here with respect to previous papers on the subject is that the right-hand side explicitly depends on the solution $u$ and we have a mismatch between the weight on the left-hand side and the right-hand side.