We consider the Dirichlet problems \(\begin{aligned} {\left\{ \begin{array}{ll} - \textrm{div} \Bigg ( \,\Big ( |\nabla u_{p}| -1 \Big )_{+}^{p-1} \displaystyle { \frac{\nabla u_{p}}{|\nabla u_{p}|} } \Bigg ) = f & \quad \text { in } B_R \qquad \\ u_{p}=0 \, & \quad \text { on } \partial {B_R}, \end{array}\right. } \end{aligned}\) where \(p > 1\) and \(B_R \subseteq \mathbb {R}^N, \, N\ge 2\) , is the open ball centered at the origin with radius \(R>0\) . Through a well-known result by Talenti (Annali di Matematica 120:159–184, 1979), we explicitly express the gradient of the solution \(u_p\) outside the set \(\{ |\nabla u_p|\le 1\}\) , 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 \(p \rightarrow 1^+\) .
