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 Morrey’s 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 $L^q$ and $W^{1,q}$ estimates for semi-stable solutions of $-\mathrm{\Delta }u=g(u)$ in a bounded domain $\mathrm{\Omega }$ of ${\mathbb{R}}^n$. These estimates lead to an $L^{2n\mathrm{/}(n-4)}(\mathrm{\Omega })$ bound for the extremal solution of $-\Delta u=\lambda f(u)$when$n\geq 5$ and the domain is convex. We recall that extremal solutions are known to be bounded in convex domains if $n\le 4$, and that their boundedness is expected —but still unkwown— for $n\le 9$.