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<n-4 $
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<n-4 $
; 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 - A, 2020, 40 (8) : 4821-4837. doi: 10.3934/dcds.2020203
References:
[1]

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

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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

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L. CaffarelliR. Kohn and L. Nirenberg, First order interpolation inequalities with weights, Compositio Math., 53 (1984), 259-275.   Google Scholar

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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

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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

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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

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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

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C.-H. HsiaC.-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

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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

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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

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E. Mitidieri, A Rellich type identity and applications, Comm. Partial Differential Equations, 18 (1993), 125-151.  doi: 10.1080/03605309308820923.  Google Scholar

[19]

E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $\mathbb{R}^n$, Differential Integral Equations, 9 (1996), 465-479.   Google Scholar

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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

[21]

J. Serrin and H. H. Zou, Non-existence of positive solutions of Lane-Emden systems, Differential Integral Equations, 9 (1996), 635-653.   Google Scholar

[22]

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

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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

show all references

References:
[1]

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

[2]

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

[3]

L. A. CaffarelliB. 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

[4]

L. CaffarelliR. Kohn and L. Nirenberg, First order interpolation inequalities with weights, Compositio Math., 53 (1984), 259-275.   Google Scholar

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

M. FazlyJ.-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

[11]

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

[12]

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

[13]

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

[14]

C.-H. HsiaC.-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

[15]

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

[16]

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

[17]

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

[18]

E. Mitidieri, A Rellich type identity and applications, Comm. Partial Differential Equations, 18 (1993), 125-151.  doi: 10.1080/03605309308820923.  Google Scholar

[19]

E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $\mathbb{R}^n$, Differential Integral Equations, 9 (1996), 465-479.   Google Scholar

[20]

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

[21]

J. Serrin and H. H. Zou, Non-existence of positive solutions of Lane-Emden systems, Differential Integral Equations, 9 (1996), 635-653.   Google Scholar

[22]

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

[23]

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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