    doi: 10.3934/cpaa.2021149
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

## 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 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, doi: 10.3934/cpaa.2021149 ##### References:   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  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  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  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  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.  Google Scholar  R. Frank and T. König, Singular solution to a semilinear biharmonic equation with a general critical nonlinearity, Rend. Lincei Mat. Appl., 30 (2019), 817-846. doi: 10.4171/RLM/871.  Google Scholar  Z. M. Guo, X. Huang, L. P. Wang and J. C. Wei, On Delaunay solutions of a biharmonic elliptic equation with critical exponent, J. Anal. Math., 140 (2020), 371-394. doi: 10.1007/s11854-020-0096-5.  Google Scholar  Z. M. Guo, F. S. Wan and L. P. Wang, Embeddings of weighted Sobolev spaces and a weighted fourth-order elliptic equation, Commun. Contemp. Math., 22 (2020), 1950057, 40 pp. doi: 10.1142/S0219199719500573.  Google Scholar  X. Huang and L. P. Wang, Classification to the positive radial solutions with weighted biharmonic equation, Discrete Contin. Dyn. Syst., 40 (2020), 4821-4837. doi: 10.3934/dcds.2020203.  Google Scholar  X. Huang and D. Ye, Hardy-Rellich type equalities, preprint. Google Scholar  C. S. Lin, A classification of solutions of a conformally invariant fourth order equation in$\mathbb{R}^N$, Comment. Math. Helv., 73 (1998), 206-231. doi: 10.1007/s000140050052.  Google Scholar  P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, I, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109–145. Google Scholar  P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, II, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 223–283. Google Scholar  R. Musina, Weighted Sobolev spaces of radially symmetric functions, Ann. Mat. Pura Appl., 193 (2014), 1626-1659. doi: 10.1007/s10231-013-0348-4.  Google Scholar show all references ##### References:   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  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  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  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  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.  Google Scholar  R. Frank and T. König, Singular solution to a semilinear biharmonic equation with a general critical nonlinearity, Rend. Lincei Mat. Appl., 30 (2019), 817-846. doi: 10.4171/RLM/871.  Google Scholar  Z. M. Guo, X. Huang, L. P. Wang and J. C. Wei, On Delaunay solutions of a biharmonic elliptic equation with critical exponent, J. Anal. Math., 140 (2020), 371-394. doi: 10.1007/s11854-020-0096-5.  Google Scholar  Z. M. Guo, F. S. Wan and L. P. Wang, Embeddings of weighted Sobolev spaces and a weighted fourth-order elliptic equation, Commun. Contemp. Math., 22 (2020), 1950057, 40 pp. doi: 10.1142/S0219199719500573.  Google Scholar  X. Huang and L. P. Wang, Classification to the positive radial solutions with weighted biharmonic equation, Discrete Contin. Dyn. Syst., 40 (2020), 4821-4837. doi: 10.3934/dcds.2020203.  Google Scholar  X. Huang and D. Ye, Hardy-Rellich type equalities, preprint. Google Scholar  C. S. Lin, A classification of solutions of a conformally invariant fourth order equation in$\mathbb{R}^N$, Comment. Math. Helv., 73 (1998), 206-231. doi: 10.1007/s000140050052.  Google Scholar  P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, I, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109–145. Google Scholar  P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, II, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 223–283. Google Scholar  R. Musina, Weighted Sobolev spaces of radially symmetric functions, Ann. Mat. Pura Appl., 193 (2014), 1626-1659. doi: 10.1007/s10231-013-0348-4.  Google Scholar   P. Lima, L. Morgado. Analysis of singular boundary value problems for an Emden-Fowler equation. Communications on Pure & Applied Analysis, 2006, 5 (2) : 321-336. doi: 10.3934/cpaa.2006.5.321  Jingbo Dou, Ye Li. Classification of extremal functions to logarithmic Hardy-Littlewood-Sobolev inequality on the upper half space. Discrete & Continuous Dynamical Systems, 2018, 38 (8) : 3939-3953. doi: 10.3934/dcds.2018171  Naoki Shioji, Kohtaro Watanabe. Uniqueness of positive radial solutions of the Brezis-Nirenberg problem on thin annular domains on$ {\mathbb S}^n \$ and symmetry breaking bifurcations. Communications on Pure & Applied Analysis, 2020, 19 (10) : 4727-4770. doi: 10.3934/cpaa.2020210  Sze-Bi Hsu, Bernold Fiedler, Hsiu-Hau Lin. Classification of potential flows under renormalization group transformation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 437-446. doi: 10.3934/dcdsb.2016.21.437  Fanni M. Sélley. Symmetry breaking in a globally coupled map of four sites. Discrete & Continuous Dynamical Systems, 2018, 38 (8) : 3707-3734. doi: 10.3934/dcds.2018161  Lucio Cadeddu, Giovanni Porru. Symmetry breaking in problems involving semilinear equations. Conference Publications, 2011, 2011 (Special) : 219-228. doi: 10.3934/proc.2011.2011.219  Hwai-Chiuan Wang. Stability and symmetry breaking of solutions of semilinear elliptic equations. Conference Publications, 2005, 2005 (Special) : 886-894. doi: 10.3934/proc.2005.2005.886  Claudia Anedda, Giovanni Porru. Symmetry breaking and other features for Eigenvalue problems. Conference Publications, 2011, 2011 (Special) : 61-70. doi: 10.3934/proc.2011.2011.61  E. García-Toraño Andrés, Bavo Langerock, Frans Cantrijn. Aspects of reduction and transformation of Lagrangian systems with symmetry. Journal of Geometric Mechanics, 2014, 6 (1) : 1-23. doi: 10.3934/jgm.2014.6.1  Sanjay Dharmavaram, Timothy J. Healey. Direct construction of symmetry-breaking directions in bifurcation problems with spherical symmetry. Discrete & Continuous Dynamical Systems - S, 2019, 12 (6) : 1669-1684. doi: 10.3934/dcdss.2019112  Boumediene Abdellaoui, Fethi Mahmoudi. An improved Hardy inequality for a nonlocal operator. Discrete & Continuous Dynamical Systems, 2016, 36 (3) : 1143-1157. doi: 10.3934/dcds.2016.36.1143  Filomena Pacella, Dora Salazar. Asymptotic behaviour of sign changing radial solutions of Lane Emden Problems in the annulus. Discrete & Continuous Dynamical Systems - S, 2014, 7 (4) : 793-805. doi: 10.3934/dcdss.2014.7.793  Linfeng Mei, Zongming Guo. Morse indices and symmetry breaking for the Gelfand equation in expanding annuli. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1509-1523. doi: 10.3934/dcdsb.2017072  Anna Goƚȩbiewska, Norimichi Hirano, Sƚawomir Rybicki. Global symmetry-breaking bifurcations of critical orbits of invariant functionals. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2005-2017. doi: 10.3934/dcdss.2019129  Ze Cheng, Congming Li. An extended discrete Hardy-Littlewood-Sobolev inequality. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 1951-1959. doi: 10.3934/dcds.2014.34.1951  Igor Freire, Ben Muatjetjeja. Symmetry analysis of a Lane-Emden-Klein-Gordon-Fock system with central symmetry. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 667-673. doi: 10.3934/dcdss.2018041  Xia Huang, Liping Wang. Classification to the positive radial solutions with weighted biharmonic equation. Discrete & Continuous Dynamical Systems, 2020, 40 (8) : 4821-4837. doi: 10.3934/dcds.2020203  Chang-Shou Lin, Lei Zhang. Classification of radial solutions to Liouville systems with singularities. Discrete & Continuous Dynamical Systems, 2014, 34 (6) : 2617-2637. doi: 10.3934/dcds.2014.34.2617  Orlando Lopes. Uniqueness and radial symmetry of minimizers for a nonlocal variational problem. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2265-2282. doi: 10.3934/cpaa.2019102  Xue Yang, Xinglong Wu. Wave breaking and persistent decay of solution to a shallow water wave equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 2149-2165. doi: 10.3934/dcdss.2016089

2020 Impact Factor: 1.916