Classification of Positive Radial Solutions to A Weighted Biharmonic Equation

TL;DR

This paper classifies positive radial solutions to a weighted biharmonic equation, establishing existence, periodicity, and symmetry properties, and relates these to inequalities of Caffarelli-Kohn-Nirenberg type.

Contribution

It provides new existence and classification results for radial solutions of a weighted fourth-order PDE, including conditions for periodicity and symmetry breaking.

Findings

Existence of radial solutions under certain parameters.

Periodic behavior of solutions depending on parameter ranges.

Results on symmetry breaking and best constants in related inequalities.

Abstract

In this paper, we consider the weighted fourth order equation $$\Delta(|x|^{-\alpha}\Delta u)+\lambda \text{div}(|x|^{-\alpha-2}\nabla u)+\mu|x|^{-\alpha-4}u=|x|^\beta u^p\quad \text{in} \quad \mathbb{R}^n \backslash \{0\},$$ where $n\geq 5$, $-n<\alpha<n-4$, $p>1$ and $(p,\alpha,\beta,n)$ belongs to the critical hyperbola $$\frac{n+\alpha}{2}+\frac{n+\beta}{p+1}=n-2.$$ We prove the existence of radial solutions to the equation for some $\lambda$ and $\mu$. On the other hand, let $v(t):=|x|^{\frac{n-4-\alpha}{2}}u(|x|)$, $t=-\ln |x|$, then for the radial solution $u$ with non-removable singularity at origin, $v(t)$ is a periodic function if $\alpha \in (-2,n-4)$ and $\lambda$, $\mu$ satisfy some conditions; while for $\alpha \in (-n,-2]$, there exists a radial solution with non-removable singularity and the corresponding function $v(t)$ is not periodic. We also get some results about the best constant and symmetry breaking, which is closely related to the Caffarelli-Kohn-Nirenberg type inequality.