$W^{1,q}$ estimates for the extremal solution of reaction-diffusion problems

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 $\lim_{t\rightarrow +\infty}f(t)/t=+\infty$.