Article Contents
Article Contents

# Classification of positive radial solutions to a weighted biharmonic equation

The author is partially supported by NSFC (No. 11431005)

• 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.

Mathematics Subject Classification: Primary: 35B09, 35B07; Secondary: 35B10, 35J30, 35J75.

 Citation:

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