# Non-radial solutions to a bi-harmonic equation with negative exponent

## TL;DR

This paper establishes the existence of non-radial entire solutions to a specific bi-harmonic equation with a negative exponent in three-dimensional space, addressing an open question in the field.

## Contribution

It proves the existence of non-radial solutions to a bi-harmonic PDE with negative exponent, solving an open problem posed by McKenna and Reichel.

## Key findings

- Existence of non-radial solutions for q>1
- Answers an open question in PDE theory
- Extends understanding of bi-harmonic equations with negative exponents

## Abstract

We prove the existence of non-radial entire solution to $$\Delta^2 u+u^{-q}=0\quad\text{in }\mathbb{R}^3,\quad u>0,$$ for $q>1$. This answers an open question raised by P. J. McKenna and W. Reichel (E. J. D. E. \textbf{37} (2003) 1-13).