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

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

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.
Citation: Zongming Guo, Xiaohong Guan, Yonggang Zhao. Uniqueness and asymptotic behavior of solutions of a biharmonic equation with supercritical exponent. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2613-2636. doi: 10.3934/dcds.2019109
References:
[1]

G. ArioliF. GazzolaH. 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. Google Scholar

[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. Google Scholar

[3]

E. BerchioA. FarinaA. 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. Google Scholar

[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. Google Scholar

[5]

J. DavilaL. DupaigneK. 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. Google Scholar

[6]

J. DavilaI 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[9]

A. FerreroH. 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. Google Scholar

[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. Google Scholar

[11]

F. GazzolaH. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[16]

Z. M. GuoX. 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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[19]

Z. M. GuoJ. 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. Google Scholar

[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. Google Scholar

[21]

H. HajlaouiA. 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. Google Scholar

[22]

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

[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. Google Scholar

[24]

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

[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. Google Scholar

[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. Google Scholar

[27]

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

[28]

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

[29]

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

[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. Google Scholar

show all references

References:
[1]

G. ArioliF. GazzolaH. 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. Google Scholar

[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. Google Scholar

[3]

E. BerchioA. FarinaA. 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. Google Scholar

[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. Google Scholar

[5]

J. DavilaL. DupaigneK. 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. Google Scholar

[6]

J. DavilaI 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[9]

A. FerreroH. 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. Google Scholar

[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. Google Scholar

[11]

F. GazzolaH. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[16]

Z. M. GuoX. 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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[19]

Z. M. GuoJ. 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. Google Scholar

[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. Google Scholar

[21]

H. HajlaouiA. 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. Google Scholar

[22]

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

[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. Google Scholar

[24]

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

[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. Google Scholar

[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. Google Scholar

[27]

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

[28]

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

[29]

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

[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. Google Scholar

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