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

## TL;DR

This paper presents a simplified, unified proof for known boundedness results of stable solutions to semilinear elliptic equations, extending some estimates to higher dimensions and addressing longstanding open problems.

## Contribution

The paper offers a new, simpler proof of existing boundedness results and introduces new estimates in higher dimensions for stable solutions.

## Key findings

- Established $L^ ext{p}$ bounds for stable solutions in dimension 5.
- Unified proof covering all dimensions up to 4 and radial case up to 9.
- Extended understanding of solution bounds in higher dimensions.

## 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 $n \leq 4$ for all $C^1$ nonlinearities $f$. In the radial case, the same is true for $n \leq 9$. 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 $5 \leq n \leq 9$ 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.