  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 $-\Delta u=g(u)$ in
a bounded domain $\Omega$ of $\mathbb{R}^n$. These estimates lead to an
$L^{2n/(n-4)}(\Omega)$ 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 $n\leq 4$, and that
their boundedness is expected ---but still unkwown--- for $n\leq 9$.
