W1,qW1,q estimates for the extremal solution of reaction–diffusion problems

We establish a new W1,2n−1n−2 estimate for the extremal solution of −Δu=λf(u)−Δu=λf(u) in a smooth bounded domain ΩΩ of RnRn, which is convex, for arbitrary positive and increasing nonlinearities f∈C1(R)f∈C1(R) satisfying limt→+∞f(t)/t=+∞limt→+∞f(t)/t=+∞.