December  2021, 29(6): 3775-3803. doi: 10.3934/era.2021061

Weighted fourth order elliptic problems in the unit ball

1. 

Department of Mathematics, Henan Normal University, Xinxiang 453007, China

2. 

School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, China

* Corresponding author: Fangshu Wan

Dedicated to Professor Norman Dancer on the occasion of his 75th birthday

Received  April 2021 Revised  July 2021 Published  December 2021 Early access  August 2021

Fund Project: The first author is supported by NSFC 11571093

Existence and uniqueness of positive radial solutions of some weighted fourth order elliptic Navier and Dirichlet problems in the unit ball $ B $ are studied. The weights can be singular at $ x = 0 \in B $. Existence of positive radial solutions of the problems is obtained via variational methods in the weighted Sobolev spaces. To obtain the uniqueness results, we need to know exactly the asymptotic behavior of the solutions at the singular point $ x = 0 $.

Citation: Zongming Guo, Fangshu Wan. Weighted fourth order elliptic problems in the unit ball. Electronic Research Archive, 2021, 29 (6) : 3775-3803. doi: 10.3934/era.2021061
References:
[1]

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

[2]

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

[3]

F. Catrina and Z.-Q. Wang, On the Caffarelli-Kohn-Nirenberg inequalities: Sharp constants, existence (and nonexistence), and symmetry of extremal functions, Comm. Pure Appl. Math., 54 (2001), 229-258.   Google Scholar

[4]

K. S. Chou and C. W. Chu, On the best constant for a weighted Sobolev-Hardy inequality, J. London Math. Soc., 48 (1993), 137-151.  doi: 10.1112/jlms/s2-48.1.137.  Google Scholar

[5]

E. N. DancerY. Du and Z. Guo, Finite Morse index solutions of an elliptic equation with supercritical exponent, J. Differential Equations, 250 (2011), 3281-3310.  doi: 10.1016/j.jde.2011.02.005.  Google Scholar

[6]

Y. Du and Z. Guo, Finite Morse index solutions of weighted elliptic equations and the critical exponents, Calc. Var. PDEs., 54 (2015), 3161-3181.  doi: 10.1007/s00526-015-0897-z.  Google Scholar

[7]

Y. DuZ. Guo and K. Wang, Monotonicity formula and $\epsilon$-regularity of stable solutions to supercritical problems and applications to finite Morse index solutions, Calc. Var. PDEs., 50 (2014), 615-638.  doi: 10.1007/s00526-013-0649-x.  Google Scholar

[8]

D. Guedes de FigueiredoE. M. dos Santos and O. H. Miyagaki, Sobolev spaces of symmetric functions and applications, J. Funct. Anal., 261 (2011), 3735-3770.  doi: 10.1016/j.jfa.2011.08.016.  Google Scholar

[9]

Z. GuoX. Guan and F. Wan, Existence and regularity of positive solutions of a degenerate elliptic problem, Math. Nachr., 292 (2019), 56-78.  doi: 10.1002/mana.201700352.  Google Scholar

[10]

Z. Guo, X. Huang and D. Ye, Existence and nonexistence results for a weighted elliptic equation in exterior domains, Z. Angew. Math. Phy., 71 (2020), Paper No. 116, 9 pp. doi: 10.1007/s00033-020-01338-0.  Google Scholar

[11]

Z. GuoJ. Li and F. Wan, Asymptotic behavior at the isolated singularities of solutions of some equations on singular manifolds with conical metrics, Comm. Partial Differential Equations, 45 (2020), 1647-1681.  doi: 10.1080/03605302.2020.1784210.  Google Scholar

[12]

Z. Guo and F. Wan, Further study of a weighted elliptic equation, Sci. China Math., 60 (2017), 2391-2406.  doi: 10.1007/s11425-017-9134-7.  Google Scholar

[13]

Z. Guo, F. Wan and L. Wang, Embeddings of weighted Sobolev spaces and a weighted fourth order elliptic equation, Comm. Contemp. Math., 22 (2020), 1950057, 40 pp. doi: 10.1142/S0219199719500573.  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), 1623-1653.  doi: 10.1512/iumj.2011.60.4376.  Google Scholar

[15]

C.-S. Lin and Z.-Q. Wang, Symmetry of extremal functions for the Caffarelli-Kohn-Nirenberg inequalities, Proc. Amer. Math. Soc., 132 (2004), 1685-1691.  doi: 10.1090/S0002-9939-04-07245-4.  Google Scholar

[16]

Q. H. Phan and P. Souplet, Liouville-type theorems and bounds of solutions for Hardy-Hénon equations, J. Differential Equations, 252 (2012), 2544-2562.  doi: 10.1016/j.jde.2011.09.022.  Google Scholar

[17]

S. Rebhi and C. Wang, Classification of finite Morse index solutions for Hénon type elliptic equation $-\Delta u=|x|^\alpha u_+^p$, Calc. Var. PDEs., 50 (2014), 847-866.  doi: 10.1007/s00526-013-0658-9.  Google Scholar

show all references

References:
[1]

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

[2]

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

[3]

F. Catrina and Z.-Q. Wang, On the Caffarelli-Kohn-Nirenberg inequalities: Sharp constants, existence (and nonexistence), and symmetry of extremal functions, Comm. Pure Appl. Math., 54 (2001), 229-258.   Google Scholar

[4]

K. S. Chou and C. W. Chu, On the best constant for a weighted Sobolev-Hardy inequality, J. London Math. Soc., 48 (1993), 137-151.  doi: 10.1112/jlms/s2-48.1.137.  Google Scholar

[5]

E. N. DancerY. Du and Z. Guo, Finite Morse index solutions of an elliptic equation with supercritical exponent, J. Differential Equations, 250 (2011), 3281-3310.  doi: 10.1016/j.jde.2011.02.005.  Google Scholar

[6]

Y. Du and Z. Guo, Finite Morse index solutions of weighted elliptic equations and the critical exponents, Calc. Var. PDEs., 54 (2015), 3161-3181.  doi: 10.1007/s00526-015-0897-z.  Google Scholar

[7]

Y. DuZ. Guo and K. Wang, Monotonicity formula and $\epsilon$-regularity of stable solutions to supercritical problems and applications to finite Morse index solutions, Calc. Var. PDEs., 50 (2014), 615-638.  doi: 10.1007/s00526-013-0649-x.  Google Scholar

[8]

D. Guedes de FigueiredoE. M. dos Santos and O. H. Miyagaki, Sobolev spaces of symmetric functions and applications, J. Funct. Anal., 261 (2011), 3735-3770.  doi: 10.1016/j.jfa.2011.08.016.  Google Scholar

[9]

Z. GuoX. Guan and F. Wan, Existence and regularity of positive solutions of a degenerate elliptic problem, Math. Nachr., 292 (2019), 56-78.  doi: 10.1002/mana.201700352.  Google Scholar

[10]

Z. Guo, X. Huang and D. Ye, Existence and nonexistence results for a weighted elliptic equation in exterior domains, Z. Angew. Math. Phy., 71 (2020), Paper No. 116, 9 pp. doi: 10.1007/s00033-020-01338-0.  Google Scholar

[11]

Z. GuoJ. Li and F. Wan, Asymptotic behavior at the isolated singularities of solutions of some equations on singular manifolds with conical metrics, Comm. Partial Differential Equations, 45 (2020), 1647-1681.  doi: 10.1080/03605302.2020.1784210.  Google Scholar

[12]

Z. Guo and F. Wan, Further study of a weighted elliptic equation, Sci. China Math., 60 (2017), 2391-2406.  doi: 10.1007/s11425-017-9134-7.  Google Scholar

[13]

Z. Guo, F. Wan and L. Wang, Embeddings of weighted Sobolev spaces and a weighted fourth order elliptic equation, Comm. Contemp. Math., 22 (2020), 1950057, 40 pp. doi: 10.1142/S0219199719500573.  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), 1623-1653.  doi: 10.1512/iumj.2011.60.4376.  Google Scholar

[15]

C.-S. Lin and Z.-Q. Wang, Symmetry of extremal functions for the Caffarelli-Kohn-Nirenberg inequalities, Proc. Amer. Math. Soc., 132 (2004), 1685-1691.  doi: 10.1090/S0002-9939-04-07245-4.  Google Scholar

[16]

Q. H. Phan and P. Souplet, Liouville-type theorems and bounds of solutions for Hardy-Hénon equations, J. Differential Equations, 252 (2012), 2544-2562.  doi: 10.1016/j.jde.2011.09.022.  Google Scholar

[17]

S. Rebhi and C. Wang, Classification of finite Morse index solutions for Hénon type elliptic equation $-\Delta u=|x|^\alpha u_+^p$, Calc. Var. PDEs., 50 (2014), 847-866.  doi: 10.1007/s00526-013-0658-9.  Google Scholar

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