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.
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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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

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Z. M. GuoX. HuangL. 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

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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

[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

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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. GuoX. HuangL. 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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