Regularity of stable solutions up to dimension 7 in domains of double revolution

Abstract

We consider the class of semi-stable positive solutions to semilinear equations $-\Delta u=f(u)$ in a bounded domain $\Omega \subset {\mathbb{R}}^n$ of double revolution, that is, a domain invariant under rotations of the first $m$ variables and of the last $n-m$ variables. We assume $2\le m\le n-2$. When the domain is convex, we establish a priori $L^p$ and $H_0^1$ bounds for each dimension $n$, with $p=\infty$ when $n\le 7$. These estimates lead to the boundedness of the extremal solution of $-\Delta u=\lambda f(u)$ in every convex domain of double revolution when $n\le 7$. The boundedness of extremal solutions is known when $n\le 3$ for any domain $\Omega$, in dimension $n=4$ when the domain is convex, and in dimensions $5\le n\le 9$ in the radial case. Except for the radial case, our result is the first partial answer valid for all nonlinearities $f$ in dimensions $5\le n\le 9$.

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