Regularity of stable solutions of $p$-Laplace equations through geometric Sobolev type inequalities

Details Regularity of stable solutions of $p$-Laplace equations through geometric Sobolev type inequalities Regularity of stable solutions of $p$-Laplace equations through geometric Sobolev type inequalities

2012-01-17

arxiv.org/abs/1201.3486v1

Mathematics

Analysis of PDEs

26 pages

Scientific paper

In this paper we prove a Sobolev and a Morrey type inequality involving the mean curvature and the tangential gradient with respect to the level sets of the function that appears in the inequalities. Then, as an application, we establish \textit{a priori} estimates for semi-stable solutions of $-\Delta_p u= g(u)$ in a smooth bounded domain $\Omega\subset \mathbb{R}^n$. In particular, we obtain new $L^r$ and $W^{1,r}$ bounds for the extremal solution $u^\star$ when the domain is strictly convex. More precisely, we prove that $u^\star\in L^\infty(\Omega)$ if $n\leq p+2$ and $u^\star\in L^{\frac{np}{n-p-2}}(\Omega)\cap W^{1,p}_0(\Omega)$ if $n>p+2$.

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