Geometric-type Sobolev inequalities and applications to the regularity of minimizers

The purpose of this paper is twofold. We first prove a weighted Sobolev inequality and part of a weighted Morreyʼs inequality, where the weights are a power of the mean curvature of the level sets of the function appearing in the inequalities. Then, as main application of our inequalities, we establish new Lq and W1,q estimates for semi-stable solutions of −Δu=g(u) in a bounded domain Ω of Rn. These estimates lead to an L2n/(n−4)(Ω) bound for the extremal solution of −Δu=λf(u) when n⩾5 and the domain is convex. We recall that extremal solutions are known to be bounded in convex domains if n⩽4, and that their boundedness is expected — but still unknown — for n⩽9.