Classification of Positive Radial Solutions to A Weighted Biharmonic Equation

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.