# Boundedness of stable solutions to semilinear elliptic equations: a survey

## TL;DR

This survey reviews known boundedness results for stable solutions to semilinear elliptic equations, highlighting current estimates, open problems in higher dimensions, and the special case of radial solutions.

## Contribution

It compiles and discusses existing $L^{
abla}$ estimates for stable solutions, emphasizing the open problem of their validity in dimensions 5 to 9.

## Key findings

- $L^{
abla}$ estimates hold up to dimension 4
- Open problem for dimensions 5 to 9
- Radial case established in higher dimensions

## Abstract

This article is a survey on boundedness results for stable solutions to semilinear elliptic problems. For these solutions, we present the currently known $L^{\infty}$ estimates that hold for all nonlinearities. Such estimates are known to hold up to dimension 4. They are expected to be true also in dimensions 5 to 9, but this is still an open problem which has only been established in the radial case.