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May  2019, 39(5): 2613-2636. doi: 10.3934/dcds.2019109

Uniqueness and asymptotic behavior of solutions of a biharmonic equation with supercritical exponent

Zongming Guo , Xiaohong Guan and  Yonggang Zhao ,

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

* Corresponding author: ygzhao@aliyun.com

Received  April 2018 Revised  August 2018 Published  January 2019

Fund Project: The first author is supported by NSF grants 11171092 and 11571093. The research of the third author is supported by Key Scientific Research Project for Colleges and Universities of Henan Province grant 18A110024, Ph.D. Research Foundation of Henan Normal University (QD16149) and Natural Science Foundation of Henan Normal University (2016QK01).

Existence and uniqueness of positive radial solution
$ u_p $
 of the Navier boundary value problem:
$ \left \{ \begin{array}{ll} \Delta^2 u = u^p \;\;\; &\mbox{in $ \mathbb{R} ^N \backslash {\overline B}$},\\ u>0 \;\;\; &\mbox{in $ \mathbb{R} ^N \backslash {\overline B}$},\\ u = \Delta u = 0 \;\;\; &\mbox{on $\partial B$}, \end{array} \right. $
where
$ B \subset \mathbb{R} ^N \; (N \geq 5) $
 is the unit ball and
$ p>\frac{N+4}{N-4} $
, are obtained. Meanwhile, the asymptotic behavior as
$ p \to \infty $
 of
$ u_p $
 is studied. We also find the conditions such that
$ u_p $
 is non-degenerate.
Keywords: Biharmonic equations, supercritical exponent, uniqueness, exterior domains, non-degenerate.
Mathematics Subject Classification: Primary: 35B45; Secondary: 35J40.
Citation: Zongming Guo, Xiaohong Guan, Yonggang Zhao. Uniqueness and asymptotic behavior of solutions of a biharmonic equation with supercritical exponent. Discrete and Continuous Dynamical Systems, 2019, 39 (5) : 2613-2636. doi: 10.3934/dcds.2019109
References:
[1]

G. Arioli, F. Gazzola, H. C. Grunau and E. Mitidieri, A semilinear fourth order elliptic problem with exponential nonlinearity, SIAM J. Math. Anal., 36 (2005), 1226-1258.  doi: 10.1137/S0036141002418534.

[2]

A. Bahri and J. M. Coron, On a nonlinear elliptic equation involving the critical Sobolev exponent: The effect of the topology of the domain, Comm. Pure Applied Math., 41 (1988), 253-294.  doi: 10.1002/cpa.3160410302.

[3]

E. Berchio, A. Farina, A. Ferrero and F. Gazzola, Existence and stability of entire solutions to a semilinear fourth order elliptic problem, J. Differential Equations, 252 (2012), 2596-2612.  doi: 10.1016/j.jde.2011.09.028.

[4]

R. Dalmasso, Uniqueness theorems for some fourth-order elliptic equations, Proc. Amer. Math. Soc., 123 (1995), 1177-1183.  doi: 10.1090/S0002-9939-1995-1242078-X.

[5]

J. Davila, L. Dupaigne, K. L. Wang and J. C. Wei, A monotonicity formula and a Liouville-type theorem for a fourth order supercritical problem, Adv. Math., 258 (2014), 240-285.  doi: 10.1016/j.aim.2014.02.034.

[6]

J. Davila, I Flores and I. Guerra, Multiplicity of solutions for a fourth order problem with power-type nonlinearity, Math. Ann., 348 (2010), 143-193.  doi: 10.1007/s00208-009-0476-8.

[7]

M. del Pino and J. C. Wei, Supercritical elliptic problems in domains with small holes, Ann. I. H. Poincaré-AN, 24 (2007), 507-520.  doi: 10.1016/j.anihpc.2006.03.001.

[8]

F. Ebobisse and M. O. Ahmedou, On a nonlinear fourth order elliptic equation involving the critical Sobolev exponent, Nonlinear Anal., 52 (2003), 1535-1552.  doi: 10.1016/S0362-546X(02)00273-0.

[9]

A. Ferrero, H. C. Grunau and P. Karageorgis, Supercritical biharmonic equations with power-type nonlinearity, Annali di Matematica, 188 (2009), 171-185.  doi: 10.1007/s10231-008-0070-9.

[10]

F. Gazzola and H. C. Grunau, Radial entire solutions for supercritical biharmonic equations, Math. Ann., 334 (2006), 905-936.  doi: 10.1007/s00208-005-0748-x.

[11]

F. Gazzola, H. C. Grunau and M. Squassina, Existence and nonexistence results for critical growth biharmonic elliptic equations, Calc. Var. PDEs, 18 (2003), 117-143.  doi: 10.1007/s00526-002-0182-9.

[12]

N. Ghoussoub and A. Moradifam, Bessel pairs and optimal Hardy and Hardy-Rellich inequalities, Math. Ann., 349 (2011), 1-57.  doi: 10.1007/s00208-010-0510-x.

[13]

N. Ghoussoub and A. Moradifam, Functional Inequalities: New Perspectives and New Applications, Mathematical Surveys and Monographs, vol. 187, American Mathematical Society, 2013. doi: 10.1090/surv/187.

[14]

M. Grossi, Asymptotic behaviour of the Kazdan-Warner solution in the annulus, J. Differential Equations, 223 (2006), 96-111.  doi: 10.1016/j.jde.2005.08.003.

[15]

Z. M. Guo, Further study of entire radial solutions of a biharmonic equation with exponential nonlinearity, Ann. di Matematica, 193 (2014), 187-201.  doi: 10.1007/s10231-012-0272-z.

[16]

Z. M. Guo, X. Huang and F. Zhou, Radial symmetry of entire solutions of a bi-harmonic equation with exponential nonlinearity, J. Funct. Anal., 268 (2015), 1972-2004.  doi: 10.1016/j.jfa.2014.12.010.

[17]

Z. M. Guo and J. C. Wei, Qualitative properties of entire radial solutions for a biharmonic equation with supercritical nonlinearity, Proc. Amer. Math. Soc., 138 (2010), 3957-3964.  doi: 10.1090/S0002-9939-10-10374-8.

[18]

Z. M. Guo, F. S. Wan and L. P. Wang, Embeddings of weighted Sobolev spaces and a weighted fourth order elliptic equation, arXiv: 1803.11298, (2018), in press.

[19]

Z. M. Guo, J. C. Wei and F. Zhou, Singular radial entire solutions and weak solutions with prescribed singular set for a biharmonic equation, J. Differential Equations, 263 (2017), 1188-1224.  doi: 10.1016/j.jde.2017.03.019.

[20]

Y. X. Guo and J. C. Wei, Supercritical biharmonic elliptic problems in domains with small holes, Math. Nachr., 282 (2009), 1724-1739.  doi: 10.1002/mana.200610814.

[21]

H. Hajlaoui, A. Harrabi and D. Ye, On stable solutions of biharmonic problem with polynomial growth, Pacific J. Math., 270 (2014), 79-93.  doi: 10.2140/pjm.2014.270.79.

[22]

P. Hartman, Ordinary Differential Equations, 2$^{nd}$ edition. Birkhäuser, Boston, 1982.

[23]

P. Karageorgis, Stability and intersection properties of solutions to the nonlinear biharmonic equation, Nonlinearity, 22 (2009), 1653-1661.  doi: 10.1088/0951-7715/22/7/009.

[24]

S. Khenissy, Nonexistence and uniqueness for hiharmonic problems with supercritical growth and domain geometry, Diff. and Integr. Equations, 24 (2011), 1093-1106. 

[25]

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.

[26]

P. J. McKenna and W. Reichel, Radial solutions of singular nonlinear biharmonic equations and applications to conformal geometry, Electron. J. Differential Equations, 37 (2003), 1-13. 

[27]

E. Mitidieri, A Rellich type identity and applications, Comm. PDEs, 18 (1993), 125-151.  doi: 10.1080/03605309308820923.

[28]

A. Moradifam, Optimal weighted Hardy-Rellich inequalities on $H^2 \cap H_0^1$, J. Lond. Math. Soc., 85 (2012), 22-40. 

[29]

R. C. A. M. Van Der Vorst, Fourth order elliptic equations with critical growth, C. R. Acad. Sci. Paris, 320 (1995), 295-299. 

[30]

J. C. Wei and D. Ye, Liouville theorems for stable solutions of biharmonic problem, Math. Ann., 356 (2013), 1599-1612.  doi: 10.1007/s00208-012-0894-x.

show all references

References:
[1]

G. Arioli, F. Gazzola, H. C. Grunau and E. Mitidieri, A semilinear fourth order elliptic problem with exponential nonlinearity, SIAM J. Math. Anal., 36 (2005), 1226-1258.  doi: 10.1137/S0036141002418534.

[2]

A. Bahri and J. M. Coron, On a nonlinear elliptic equation involving the critical Sobolev exponent: The effect of the topology of the domain, Comm. Pure Applied Math., 41 (1988), 253-294.  doi: 10.1002/cpa.3160410302.

[3]

E. Berchio, A. Farina, A. Ferrero and F. Gazzola, Existence and stability of entire solutions to a semilinear fourth order elliptic problem, J. Differential Equations, 252 (2012), 2596-2612.  doi: 10.1016/j.jde.2011.09.028.

[4]

R. Dalmasso, Uniqueness theorems for some fourth-order elliptic equations, Proc. Amer. Math. Soc., 123 (1995), 1177-1183.  doi: 10.1090/S0002-9939-1995-1242078-X.

[5]

J. Davila, L. Dupaigne, K. L. Wang and J. C. Wei, A monotonicity formula and a Liouville-type theorem for a fourth order supercritical problem, Adv. Math., 258 (2014), 240-285.  doi: 10.1016/j.aim.2014.02.034.

[6]

J. Davila, I Flores and I. Guerra, Multiplicity of solutions for a fourth order problem with power-type nonlinearity, Math. Ann., 348 (2010), 143-193.  doi: 10.1007/s00208-009-0476-8.

[7]

M. del Pino and J. C. Wei, Supercritical elliptic problems in domains with small holes, Ann. I. H. Poincaré-AN, 24 (2007), 507-520.  doi: 10.1016/j.anihpc.2006.03.001.

[8]

F. Ebobisse and M. O. Ahmedou, On a nonlinear fourth order elliptic equation involving the critical Sobolev exponent, Nonlinear Anal., 52 (2003), 1535-1552.  doi: 10.1016/S0362-546X(02)00273-0.

[9]

A. Ferrero, H. C. Grunau and P. Karageorgis, Supercritical biharmonic equations with power-type nonlinearity, Annali di Matematica, 188 (2009), 171-185.  doi: 10.1007/s10231-008-0070-9.

[10]

F. Gazzola and H. C. Grunau, Radial entire solutions for supercritical biharmonic equations, Math. Ann., 334 (2006), 905-936.  doi: 10.1007/s00208-005-0748-x.

[11]

F. Gazzola, H. C. Grunau and M. Squassina, Existence and nonexistence results for critical growth biharmonic elliptic equations, Calc. Var. PDEs, 18 (2003), 117-143.  doi: 10.1007/s00526-002-0182-9.

[12]

N. Ghoussoub and A. Moradifam, Bessel pairs and optimal Hardy and Hardy-Rellich inequalities, Math. Ann., 349 (2011), 1-57.  doi: 10.1007/s00208-010-0510-x.

[13]

N. Ghoussoub and A. Moradifam, Functional Inequalities: New Perspectives and New Applications, Mathematical Surveys and Monographs, vol. 187, American Mathematical Society, 2013. doi: 10.1090/surv/187.

[14]

M. Grossi, Asymptotic behaviour of the Kazdan-Warner solution in the annulus, J. Differential Equations, 223 (2006), 96-111.  doi: 10.1016/j.jde.2005.08.003.

[15]

Z. M. Guo, Further study of entire radial solutions of a biharmonic equation with exponential nonlinearity, Ann. di Matematica, 193 (2014), 187-201.  doi: 10.1007/s10231-012-0272-z.

[16]

Z. M. Guo, X. Huang and F. Zhou, Radial symmetry of entire solutions of a bi-harmonic equation with exponential nonlinearity, J. Funct. Anal., 268 (2015), 1972-2004.  doi: 10.1016/j.jfa.2014.12.010.

[17]

Z. M. Guo and J. C. Wei, Qualitative properties of entire radial solutions for a biharmonic equation with supercritical nonlinearity, Proc. Amer. Math. Soc., 138 (2010), 3957-3964.  doi: 10.1090/S0002-9939-10-10374-8.

[18]

Z. M. Guo, F. S. Wan and L. P. Wang, Embeddings of weighted Sobolev spaces and a weighted fourth order elliptic equation, arXiv: 1803.11298, (2018), in press.

[19]

Z. M. Guo, J. C. Wei and F. Zhou, Singular radial entire solutions and weak solutions with prescribed singular set for a biharmonic equation, J. Differential Equations, 263 (2017), 1188-1224.  doi: 10.1016/j.jde.2017.03.019.

[20]

Y. X. Guo and J. C. Wei, Supercritical biharmonic elliptic problems in domains with small holes, Math. Nachr., 282 (2009), 1724-1739.  doi: 10.1002/mana.200610814.

[21]

H. Hajlaoui, A. Harrabi and D. Ye, On stable solutions of biharmonic problem with polynomial growth, Pacific J. Math., 270 (2014), 79-93.  doi: 10.2140/pjm.2014.270.79.

[22]

P. Hartman, Ordinary Differential Equations, 2$^{nd}$ edition. Birkhäuser, Boston, 1982.

[23]

P. Karageorgis, Stability and intersection properties of solutions to the nonlinear biharmonic equation, Nonlinearity, 22 (2009), 1653-1661.  doi: 10.1088/0951-7715/22/7/009.

[24]

S. Khenissy, Nonexistence and uniqueness for hiharmonic problems with supercritical growth and domain geometry, Diff. and Integr. Equations, 24 (2011), 1093-1106. 

[25]

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.

[26]

P. J. McKenna and W. Reichel, Radial solutions of singular nonlinear biharmonic equations and applications to conformal geometry, Electron. J. Differential Equations, 37 (2003), 1-13. 

[27]

E. Mitidieri, A Rellich type identity and applications, Comm. PDEs, 18 (1993), 125-151.  doi: 10.1080/03605309308820923.

[28]

A. Moradifam, Optimal weighted Hardy-Rellich inequalities on $H^2 \cap H_0^1$, J. Lond. Math. Soc., 85 (2012), 22-40. 

[29]

R. C. A. M. Van Der Vorst, Fourth order elliptic equations with critical growth, C. R. Acad. Sci. Paris, 320 (1995), 295-299. 

[30]

J. C. Wei and D. Ye, Liouville theorems for stable solutions of biharmonic problem, Math. Ann., 356 (2013), 1599-1612.  doi: 10.1007/s00208-012-0894-x.

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