  We establish a new $W^{1,2\frac{n-1}{n-2}}$ estimate for the extremal
solution of $-\Delta u=\lambda f(u)$ in a smooth bounded domain $\Omega$ of
$\mathbb{R}^n$, which is convex, for arbitrary positive and increasing
nonlinearities $f\in C^1(\mathbb{R})$ satisfying $\lim_{t\rightarrow
+\infty}f(t)/t=+\infty$.
