# American Institute of Mathematical Sciences

December  2021, 20(12): 4139-4154. doi: 10.3934/cpaa.2021149

## Classification of positive radial solutions to a weighted biharmonic equation

 School of Mathematical Sciences, East China Normal University, Shanghai 200241, China

Received  May 2021 Revised  July 2021 Published  December 2021 Early access  September 2021

Fund Project: 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 , $ 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. Citation: Yuhao Yan. Classification of positive radial solutions to a weighted biharmonic equation. Communications on Pure & Applied Analysis, 2021, 20 (12) : 4139-4154. doi: 10.3934/cpaa.2021149 ##### References:  [1] M. Bhakta and R. Musina, Entire solutions for a class of variational problems involving the biharmonic operator and Rellich potentials, Nonlinear Anal., 75 (2012), 3836-3848. doi: 10.1016/j.na.2012.02.005. Google Scholar [2] P. Caldiroli and G. Cora, Entire solutions for a class of fourth-order semilinear elliptic equations with weights, Mediterr. J. Math., 13 (2016), 657-675. doi: 10.1007/s00009-015-0519-1. Google Scholar [3] P. Caldiroli and R. Musina, On Caffarelli-Kohn-Nirenberg type inequalities for the weighted biharmonic operator in cones, Milan J. Math., 79 (2011), 657-687. doi: 10.1007/s00032-011-0167-2. Google Scholar [4] P. Caldiroli and R. Musina, Rellich inequalities with weights, Calc. Var. Partial Differ. Equ., 45 (2012), 147-164. doi: 10.1007/s00526-011-0454-3. Google Scholar [5] R. Frank and T. 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Musina, Entire solutions for a class of variational problems involving the biharmonic operator and Rellich potentials, Nonlinear Anal., 75 (2012), 3836-3848. doi: 10.1016/j.na.2012.02.005. Google Scholar [2] P. Caldiroli and G. Cora, Entire solutions for a class of fourth-order semilinear elliptic equations with weights, Mediterr. J. Math., 13 (2016), 657-675. doi: 10.1007/s00009-015-0519-1. Google Scholar [3] P. Caldiroli and R. Musina, On Caffarelli-Kohn-Nirenberg type inequalities for the weighted biharmonic operator in cones, Milan J. Math., 79 (2011), 657-687. doi: 10.1007/s00032-011-0167-2. Google Scholar [4] P. Caldiroli and R. Musina, Rellich inequalities with weights, Calc. Var. Partial Differ. Equ., 45 (2012), 147-164. doi: 10.1007/s00526-011-0454-3. Google Scholar [5] R. Frank and T. König, Classification of positive solutions to a nonlinear biharmonic equation with critical exponent, Anal. Partial Differ. Equ., 12 (2019), 1101-1113. doi: 10.2140/apde.2019.12.1101. 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