Classification of Positive Radial Solutions to A Weighted Biharmonic Equation

Abstract

In this paper, we consider the weighted fourth order equation $$\Delta (|x{|}^{-\alpha }\Delta u)+\lambda \text{div}(|x{|}^{-\alpha -2}\nabla u)+\mu |x{|}^{-\alpha -4}u=|x{|}^{\beta }{u}^p\text{in}{\mathbb{R}}^n\backslash \{0\},$$ where $n\ge 5$, $-n<\alpha <n-4$, $p>1$ 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.

arXiv comment15 pages