  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 $p \to 1^+$.
