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A generalization of Kátai's orthogonality criterion with applications
Uniqueness and asymptotic behavior of solutions of a biharmonic equation with supercritical exponent
Department of Mathematics, Henan Normal University, Xinxiang 453007, China |
$ u_p $ |
$ \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. $ |
$ B \subset \mathbb{R} ^N \; (N \geq 5) $ |
$ p>\frac{N+4}{N-4} $ |
$ p \to \infty $ |
$ u_p $ |
$ u_p $ |
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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