  This paper is devoted to the study of semi-stable radial solutions $u\in
H^1(B_1)$ of $-\Delta u=g(u) {in} B_1\setminus \{0\}$, where $g\in C^1(R)$ is a
general nonlinearity and $B_1$ is the unit ball of $R^N$. We establish sharp
pointwise estimates for such solutions. As an application of these results, we
obtain optimal pointwise estimates for the extremal solution and its
derivatives (up to order three) of the semilinear elliptic equation $-\Delta
u=\lambda f(u)$, posed in $B_1$, with Dirichlet data $u|_{\partial B_1}=0$, and
a continuous, positive, nondecreasing and convex function $f$ on $[0,\infty)$
such that $f(s)/s\to\infty$ as $s\to\infty$.
