# Singular solutions to a semilinear biharmonic equation with a general critical nonlinearity

## TL;DR

This paper classifies positive solutions with singularities of a semilinear biharmonic equation involving a general critical nonlinearity, revealing a periodic structure of solutions in logarithmic scale.

## Contribution

It introduces a classification of singular solutions for a biharmonic equation with general critical nonlinearity, showing their periodic nature in logarithmic variables.

## Key findings

- Solutions exhibit periodicity in log-scale of |x|
- Complete classification of singular solutions
- Conditions on nonlinearity g ensure solution structure

## Abstract

We consider positive solutions $u$ of the semilinear biharmonic equation $\Delta^2 u = |x|^{-\frac{n+4}{2}} g(|x|^\frac{n-4}{2} u)$ in $\mathbb R^n \setminus \{0\}$ with non-removable singularities at the origin. Under natural assumptions on the nonlinearity $g$, we show that $|x|^\frac{n-4}{2} u$ is a periodic function of $\ln |x|$ and we classify all such solutions.