    August  2020, 40(8): 4821-4837. doi: 10.3934/dcds.2020203

## Classification to the positive radial solutions with weighted biharmonic equation

 School of Mathematical Sciences, Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice, East China Normal University, Shanghai 200241, China

* Corresponding author: Liping Wang

Received  August 2019 Revised  March 2020 Published  May 2020

Fund Project: The first author is supported by NSFC 11701181 and the second author is supported by NSFC 11671144

In this paper, we consider the weighted problem
 $\Delta (|x|^{-\alpha} \Delta u) = |x|^{\beta} u^p, \qquad u(x)>0, \qquad u(x) = u(|x|)\qquad \text{in}\; \; \mathbb{R}^n\backslash{\{0}\},$
where
 $n\ge 5, -n<\alpha and $ (p, \alpha,\beta, n), p>1 $belongs to the critical hyperbola $ \frac{n+\alpha}{2}+\frac{n+\beta}{p+1} = n-2. $We give two type-homoclinic functions $ v(t): = |x|^{\frac{n-4-\alpha}{2}}u(|x|), t = -\ln |x| $. On the other hand, for radial solution $ u $with non-removable singularity at origin, $ v(t) $is periodic and classification for all periodic functions are obtained with $ -2<\alpha
; while for
 $-n<\alpha \le -2,$
there always exists a solution
 $u(|x|)$
with non-removable singularity and the corresponding function
 $v(t)$
is not periodic. It is also closely related to the Caffarelli-Kohn-Nirenberg inequality, and we get some results such as the best embedding constants and the existence in radial case. In particular, for
 $\alpha = \beta = 0$
, it is related to the
 $Q$
-curvature problem in conformal geometry.
Citation: 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
##### References:
  M. Bhakta and R. Musina, Entire solutions for a class of variational problems involving the biharmonic operator and Rellich potentials, Nonlinear Analysis, 75 (2012), 3836-3848.  doi: 10.1016/j.na.2012.02.005.  Google Scholar  M. F. Bidaut-Veron and H. Giacomini, A new dynamical approach of Emden-Fowler equations and systems, Adv. Differential Equations, 15 (2010), 1033-1082. Google Scholar  L. A. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math., 42 (1989), 271-297.  doi: 10.1002/cpa.3160420304.  Google Scholar  L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequalities with weights, Compositio Math., 53 (1984), 259-275. 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  A. Carioli and R. Musina, The homogeneous Hénon-Lane-Emden system, NoDEA Nonlinear Differential Equations Appl., 22 (2015), 1445-1459.  doi: 10.1007/s00030-015-0330-5.  Google Scholar  W. X. Chen and C. M. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622.  doi: 10.1215/S0012-7094-91-06325-8.  Google Scholar  M. Fazly, Liouville theorems for the polyharmonic Hénon-Lane-Emden system, Methods Appl. Anal., 21 (2014), 265-281.  doi: 10.4310/MAA.2014.v21.n2.a5.  Google Scholar  M. Fazly and N. Ghoussoub, On the Hénon-Lane-Emden conjecture, Discrete Contin. Dyn. Syst., 34 (2014), 2513-2533.  doi: 10.3934/dcds.2014.34.2513.  Google Scholar  M. Fazly, J.-C. Wei and X. W. Xu, A pointwise inequality for the fourth-order Lane-Emden equation, Anal. PDE, 8 (2015), 1541-1563.  doi: 10.2140/apde.2015.8.1541.  Google Scholar  R. L. Frank and T. König, Classification of positive solutions to a nonlinear biharmonic equation with critical exponent, Anal. PDE, 12 (2019), 1101-1113.  doi: 10.2140/apde.2019.12.1101.  Google Scholar  R. L. Frank and T. König, Singular solutions 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, Accepted by Journal d'Analysise Mathematique, Available from: http://www.math.ubc.ca/ jcwei/GHWW-2017-08-15.pdf. Google Scholar  C.-H. Hsia, C.-S. Lin and Z.-Q. Wang, Asymptotic symmetry and local behaviors of solutions to a class of anisotropic elliptic equations, Indiana Univ. Math. J., 60 (2011), 1623-1654.  doi: 10.1512/iumj.2011.60.4376.  Google Scholar  K. Li and Z. T. Zhang, Proof of the Hénon-Lane-Emden conjecture in $R^3$, J. Differential Equations, 266 (2019), 202-226.  doi: 10.1016/j.jde.2018.07.036.  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 limit case. I, Rev. Mat. Iberoam., 1 (1985), 145-201.   Google Scholar  E. Mitidieri, A Rellich type identity and applications, Comm. Partial Differential Equations, 18 (1993), 125-151.  doi: 10.1080/03605309308820923.  Google Scholar  E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $\mathbb{R}^n$, Differential Integral Equations, 9 (1996), 465-479. Google Scholar  É. Mitidieri and S. I. Pokhozhaev, A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities, Tr. Mat. Inst. Steklova, 234 (2001), 1-384. Google Scholar  J. Serrin and H. H. Zou, Non-existence of positive solutions of Lane-Emden systems, Differential Integral Equations, 9 (1996), 635-653. Google Scholar  M. Struwe, Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Fourth edition, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 34. Springer-Verlag, Berlin, 2008. Google Scholar  J. C. Wei and X. W. Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann., 313 (1999), 207-228.  doi: 10.1007/s002080050258.  Google Scholar

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##### References:
  M. Bhakta and R. Musina, Entire solutions for a class of variational problems involving the biharmonic operator and Rellich potentials, Nonlinear Analysis, 75 (2012), 3836-3848.  doi: 10.1016/j.na.2012.02.005.  Google Scholar  M. F. Bidaut-Veron and H. Giacomini, A new dynamical approach of Emden-Fowler equations and systems, Adv. Differential Equations, 15 (2010), 1033-1082. Google Scholar  L. A. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math., 42 (1989), 271-297.  doi: 10.1002/cpa.3160420304.  Google Scholar  L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequalities with weights, Compositio Math., 53 (1984), 259-275. 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  A. Carioli and R. Musina, The homogeneous Hénon-Lane-Emden system, NoDEA Nonlinear Differential Equations Appl., 22 (2015), 1445-1459.  doi: 10.1007/s00030-015-0330-5.  Google Scholar  W. X. Chen and C. M. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622.  doi: 10.1215/S0012-7094-91-06325-8.  Google Scholar  M. Fazly, Liouville theorems for the polyharmonic Hénon-Lane-Emden system, Methods Appl. Anal., 21 (2014), 265-281.  doi: 10.4310/MAA.2014.v21.n2.a5.  Google Scholar  M. Fazly and N. Ghoussoub, On the Hénon-Lane-Emden conjecture, Discrete Contin. Dyn. Syst., 34 (2014), 2513-2533.  doi: 10.3934/dcds.2014.34.2513.  Google Scholar  M. Fazly, J.-C. Wei and X. W. Xu, A pointwise inequality for the fourth-order Lane-Emden equation, Anal. PDE, 8 (2015), 1541-1563.  doi: 10.2140/apde.2015.8.1541.  Google Scholar  R. L. Frank and T. König, Classification of positive solutions to a nonlinear biharmonic equation with critical exponent, Anal. PDE, 12 (2019), 1101-1113.  doi: 10.2140/apde.2019.12.1101.  Google Scholar  R. L. Frank and T. König, Singular solutions 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, Accepted by Journal d'Analysise Mathematique, Available from: http://www.math.ubc.ca/ jcwei/GHWW-2017-08-15.pdf. Google Scholar  C.-H. Hsia, C.-S. Lin and Z.-Q. Wang, Asymptotic symmetry and local behaviors of solutions to a class of anisotropic elliptic equations, Indiana Univ. Math. J., 60 (2011), 1623-1654.  doi: 10.1512/iumj.2011.60.4376.  Google Scholar  K. Li and Z. T. Zhang, Proof of the Hénon-Lane-Emden conjecture in $R^3$, J. Differential Equations, 266 (2019), 202-226.  doi: 10.1016/j.jde.2018.07.036.  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 limit case. I, Rev. Mat. Iberoam., 1 (1985), 145-201.   Google Scholar  E. Mitidieri, A Rellich type identity and applications, Comm. Partial Differential Equations, 18 (1993), 125-151.  doi: 10.1080/03605309308820923.  Google Scholar  E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $\mathbb{R}^n$, Differential Integral Equations, 9 (1996), 465-479. Google Scholar  É. Mitidieri and S. I. Pokhozhaev, A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities, Tr. Mat. Inst. Steklova, 234 (2001), 1-384. Google Scholar  J. Serrin and H. H. Zou, Non-existence of positive solutions of Lane-Emden systems, Differential Integral Equations, 9 (1996), 635-653. Google Scholar  M. Struwe, Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Fourth edition, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 34. Springer-Verlag, Berlin, 2008. Google Scholar  J. C. Wei and X. W. Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann., 313 (1999), 207-228.  doi: 10.1007/s002080050258.  Google Scholar

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