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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

Yuhao Yan

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 & 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

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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

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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

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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

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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

[10]

X. Huang and D. Ye, Hardy-Rellich type equalities, preprint. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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

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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:
[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. 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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

X. Huang and D. Ye, Hardy-Rellich type equalities, preprint. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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