Classification of positive radial solutions to a weighted biharmonic equation

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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<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.

Keywords: Radial solution, classification, Emden-Fowler transformation, hardy inequality, symmetry breaking.

Mathematics Subject Classification: Primary: 35B09, 35B07; Secondary: 35B10, 35J30, 35J75.

Citation: Yuhao Yan. Classification of positive radial solutions to a weighted biharmonic equation. Communications on Pure and 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.
[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.
[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.
[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.
[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.
[6]	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.
[7]	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.
[8]	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.
[9]	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.
[10]	X. Huang and D. Ye, Hardy-Rellich type equalities, preprint.
[11]	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.
[12]	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.
[13]	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.
[14]	R. Musina, Weighted Sobolev spaces of radially symmetric functions, Ann. Mat. Pura Appl., 193 (2014), 1626-1659. doi: 10.1007/s10231-013-0348-4.

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 Anal., 75 (2012), 3836-3848. doi: 10.1016/j.na.2012.02.005.
[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.
[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.
[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.
[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.
[6]	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.
[7]	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.
[8]	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.
[9]	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.
[10]	X. Huang and D. Ye, Hardy-Rellich type equalities, preprint.
[11]	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.
[12]	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.
[13]	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.
[14]	R. Musina, Weighted Sobolev spaces of radially symmetric functions, Ann. Mat. Pura Appl., 193 (2014), 1626-1659. doi: 10.1007/s10231-013-0348-4.

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