# Symmetry of large solutions for semilinear elliptic equations in a ball

## TL;DR

This paper proves that solutions to a boundary blow-up problem for semilinear elliptic equations in a ball are radially symmetric and increasing, under certain conditions on the nonlinearity, extending previous results.

## Contribution

It establishes sharp conditions on the nonlinearity that guarantee symmetry and monotonicity of large solutions, generalizing earlier findings.

## Key findings

- Solutions are radially symmetric and increasing.
- Conditions on the asymptotic behavior of f are sufficient.
- Results extend previous symmetry theorems.

## Abstract

In this work we consider the boundary blow-up problem $$ \left\{ \begin{array}{ll} \Delta u = f(u) & \hbox{in } B\\ \ \ u=+\infty & \hbox{on }\partial B \end{array} \right. $$ where $B$ stands for the unit ball of $\mathbb{R}^N$ and $f$ is a locally Lipschitz function which is positive for large values and verifies the Keller-Osserman condition. Under an additional hypothesis on the asymptotic behavior of $f$ we show that all solutions of the above problem are radially symmetric and radially increasing. Our condition is sharp enough to generalize several results in previous literature.