On widely degenerate p-Laplace equations with symmetric data

We consider the Dirichlet problems \begin{equation*}\label{1} \begin{cases} - \mathrm{div} \Bigg( \,\Big( \abs{\nabla u_{p}} -1 \Big)_{+}^{p-1} \displaystyle{ \frac{\nabla u_{p}}{\abs{\nabla u_{p}}} } \Bigg) = f \qquad \text{ in } B_R \qquad\\ % - \sum_{i=1}^{N} \frac{\partial}{\partial x_i} a_i(x,u,\nabla u)= H(x,u) u_{p}=0 \, \hspace{14em} \text{ on } \partial{B_R}, \end{cases} \end{equation*} where $p > 1$ and $B_R \subseteq \mathbb{R}^N, \, N\geq 2$, is the open ball centered at the origin with radius $R>0$ . %of $\mathbb{R}^N \text{ with } \,N \geq 2$ and $p > 1$. \\ %and $( \,\, \cdot \,\, )_+$ stands for the positive part.\\ Through a well-known result by Talenti \cite{Tal}, we explicitly express the gradient of the solution $u_p$ outside the set $\{ \abs{\nabla u_p}\leq 1\}$, if the datum $f$ is a non-negative integrable radially decreasing function. This allows us to establish some sharp higher regularity results for the weak solutions, assuming that the datum $f$ belongs to a suitable Lorentz space, i.e. under a weaker assumption on the datum with respect to the available literature. Moreover we analyze the behaviour of $u_p$ as $p \to 1^+$.