A new proof of the boundedness results for stable solutions to semilinear elliptic equations

Abstract

We consider the class of stable solutions to semilinear equations $-\Delta u=f(u)$ in a bounded smooth domain of ${\mathbb{R}}^n$. Since 2010 an interior a priori $L^{\infty }$ bound for stable solutions is known to hold in dimensions $n\le 4$ for all $C^1$ nonlinearities $f$. In the radial case, the same is true for $n\le 9$. Here we provide with a new, simpler, and unified proof of these results. It establishes, in addition, some new estimates in higher dimensions ---for instance $L^p$ bounds for every finite~$p$ in dimension 5. Since the mid nineties, the existence of an $L^{\infty }$ bound holding for all $C^1$ nonlinearities when $5\le n\le 9$ was a challenging open problem. This has been recently solved by A. Figalli, X. Ros-Oton, J. Serra, and the author, for nonnegative nonlinearities, in a forthcoming paper.

arXiv commentTo appear in a special volume of DCDS dedicated to Luis Caffarelli. While version 1 was written approximately one year ago, version 2 contains updated bibliography and comments to an important recent result. Some few additional comments added in version 3