On widely degenerate \textit{p}-Laplace equations with symmetric data

In this paper, we consider the Dirichlet problems with a widely degenerate equation. Through a well-known result by Talenti, we explicitly express the gradient of the solution $u_p$ outside the ball with a radius of $1$, if the datum $f$ is a non-negative 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 $p \to 1^+$.