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

Abstract

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 ${\lim }_{t\to +\infty }f(t)/t=+\infty$.

10 pages

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