May  2020, 19(5): 2919-2948. doi: 10.3934/cpaa.2020128

Bôcher-type results for the fourth and higher order equations on singular manifolds with conical metrics

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

Received  October 2019 Revised  October 2019 Published  March 2020

Fund Project: The author is supported by NSFC, No. 11721101

We obtain the Bôcher-type theorems and present the sharp characterization of the asymptotic behavior at the isolated singularities of solutions of some fourth and higher order equations on singular manifolds with conical metrics. It is seen that the equations on singular manifolds with conical metrics are equivalent to weighted elliptic equations in $ B \backslash \{0\} $, where $ B \subset \mathbb{R}^N $ is the unit ball. The weights can be singular at $ x = 0 $. We present the sharp asymptotic behavior of nonnegative solutions of the weighted elliptic equations near $ x = 0 $ and the Liouville-type results for the degenerate elliptic equations in $ \mathbb{R}^N \backslash \{0\} $.

Citation: Fangshu Wan. Bôcher-type results for the fourth and higher order equations on singular manifolds with conical metrics. Communications on Pure & Applied Analysis, 2020, 19 (5) : 2919-2948. doi: 10.3934/cpaa.2020128
References:
[1]

A. ArapostathisM. K. Ghosh and S. I. Marcus, Harnack's inequality for cooperative weakly coupled elliptic systems, Commun. Partial Differ. Equ., 24 (1999), 1555-1571.  doi: 10.1080/03605309908821475.  Google Scholar

[2]

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

[3]

S. N. ArmstrongB. Sirakov and C. K. Smart, Fundamental solutions of homogeneous fully nonlinear elliptic equations, Commun. Pure Appl. Math., 64 (2011), 737-777.  doi: 10.1002/cpa.20360.  Google Scholar

[4]

E. BerchioA. FarinaA. Ferrero and F. Gazzola, Existence and stability of entire solutions to a semilinear fourth order elliptic problem, J. Differ. Equ., 252 (2012), 2596-2612.  doi: 10.1016/j.jde.2011.09.028.  Google Scholar

[5]

M. Bôcher, Singular points of functions which satisfy partial differential equations of the elliptic type, Bull. Amer. Math. Soc., 9 (1903), 455-465.  doi: 10.1090/S0002-9904-1903-01017-9.  Google Scholar

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

[7]

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

[8]

F. Catrina and Z. Q. Wang, On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions, Commun. Pure Appl. Math., 54 (2001), 229-258.  doi: 10.1002/1097-0312(200102)54:2<229::AID-CPA4>3.0.CO;2-I.  Google Scholar

[9]

C. CowanP. EspositoN. Ghoussoub and A. Moradifam, The critical dimension for a fourth order elliptic problem with singular nonlinearity, Arch. Ration. Mech. Anal., 198 (2010), 763-787.  doi: 10.1007/s00205-010-0367-x.  Google Scholar

[10]

J. DavilaL. DupaigneI. Guerra and M. Montenegro, Stable solutions for the bilaplacian with exponential nonlinearity, SIAM J. Math. Anal., 39 (2007), 565-592.  doi: 10.1137/060665579.  Google Scholar

[11]

L. DupaigneM. GherguO. Goubet and G. Warnault, The Gel'fand problem for the biharmonic operator, Arch. Ration. Mech. Anal., 208 (2013), 725-752.  doi: 10.1007/s00205-013-0613-0.  Google Scholar

[12]

D. E. EdmundsD. Fortunato and E. Jannelli, Critical exponents, Critical dimensions and the biharmonic operator, Arch. Ration. Mech. Anal., 112 (1990), 269-289.  doi: 10.1007/BF00381236.  Google Scholar

[13]

R. L. 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

[14]

P. L. Felmer and A. Quaas, Fundamental solutions and two properties of elliptic maximal and minimal operators, Trans. Amer. Math. Soc., 361 (2009), 5721-5736.  doi: 10.1090/S0002-9947-09-04566-8.  Google Scholar

[15]

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

[16]

D. Gilbarg and J. Serrin, On isolated singularites of solutions of second order elliptic differential equations, J. Anal. Math., 4 (1955/56), 309-340.  doi: 10.1007/BF02787726.  Google Scholar

[17]

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

[18]

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

[19]

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

[20]

Z. M. GuoX. H. Guan and F. S. Wan, Sobolev type embedding and weak solutions with a prescribed singular set, Sci. China Math., 59 (2016), 1975-1994.  doi: 10.1007/s11425-015-0698-0.  Google Scholar

[21]

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

[22]

Z. M. GuoX. H. Guan and Y. G. Zhao, Uniqueness and asymptotic behavior of solutions of a biharmonic equation with supercritical exponent, Discrete Conti. Dyn. Syst., 39 (2019), 2613-2636.  doi: 10.3934/dcds.2019109.  Google Scholar

[23]

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

[24]

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

[25]

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

[26]

Z. M. Guo and J. C. Wei, On a fourth order elliptic problem with negative exponent, SIAM J. Math. Anal., 40 (2008/09), 2034-2054.  doi: 10.1137/070703375.  Google Scholar

[27]

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

[28]

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

[29]

D. A. Labutin, Isolated singularities for fully nonlinear elliptic equations, J. Differ. Equ., 177 (2001), 49-76.  doi: 10.1006/jdeq.2001.3998.  Google Scholar

[30]

J. Y. Li and F. S. Wan, Bôcher-type theorem on $N$-dimensioanl manifolds with conical metric, Proc. Amer. Math. Soc., 147 (2019), 4527-4538.  doi: 10.1090/proc/14554.  Google Scholar

[31]

Y. Y. Li and L. Nguyen, Harnack inequalities and Bôcher-type theorems for conformally invariant, fully nonlinear degenerate elliptic equations, Comm. Pure Appl. Math., 67 (2014), 1843-1876.  doi: 10.1002/cpa.21502.  Google Scholar

[32]

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

[33]

C. S. Lin and Z. Q. Wang, Symmetry of extremal functions for the Caffarelli-Kohn-Nirenberg inequalities, Proc. Amer. Math. Soc., 132 (2006), 1685-1691.  doi: 10.1090/S0002-9939-04-07245-4.  Google Scholar

[34]

R. Musina, Weighted Sobolev spaces of radially symmetric functions, Ann. Mat. Pura Appl., 193 (2014), 1629-1659.  doi: 10.1007/s10231-013-0348-4.  Google Scholar

[35]

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

[36]

C. Wang and D. Ye, Some Liouville theorems for Hénon type elliptic equations, J. Funct. Anal., 262 (2012), 1705-1727.  doi: 10.1016/j.jfa.2011.11.017.  Google Scholar

[37]

J. C. Wei and X. W. Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann., 313 (1999), 207-228.  doi: 10.1007/s002080050258.  Google Scholar

show all references

References:
[1]

A. ArapostathisM. K. Ghosh and S. I. Marcus, Harnack's inequality for cooperative weakly coupled elliptic systems, Commun. Partial Differ. Equ., 24 (1999), 1555-1571.  doi: 10.1080/03605309908821475.  Google Scholar

[2]

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

[3]

S. N. ArmstrongB. Sirakov and C. K. Smart, Fundamental solutions of homogeneous fully nonlinear elliptic equations, Commun. Pure Appl. Math., 64 (2011), 737-777.  doi: 10.1002/cpa.20360.  Google Scholar

[4]

E. BerchioA. FarinaA. Ferrero and F. Gazzola, Existence and stability of entire solutions to a semilinear fourth order elliptic problem, J. Differ. Equ., 252 (2012), 2596-2612.  doi: 10.1016/j.jde.2011.09.028.  Google Scholar

[5]

M. Bôcher, Singular points of functions which satisfy partial differential equations of the elliptic type, Bull. Amer. Math. Soc., 9 (1903), 455-465.  doi: 10.1090/S0002-9904-1903-01017-9.  Google Scholar

[6]

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

[7]

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

[8]

F. Catrina and Z. Q. Wang, On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions, Commun. Pure Appl. Math., 54 (2001), 229-258.  doi: 10.1002/1097-0312(200102)54:2<229::AID-CPA4>3.0.CO;2-I.  Google Scholar

[9]

C. CowanP. EspositoN. Ghoussoub and A. Moradifam, The critical dimension for a fourth order elliptic problem with singular nonlinearity, Arch. Ration. Mech. Anal., 198 (2010), 763-787.  doi: 10.1007/s00205-010-0367-x.  Google Scholar

[10]

J. DavilaL. DupaigneI. Guerra and M. Montenegro, Stable solutions for the bilaplacian with exponential nonlinearity, SIAM J. Math. Anal., 39 (2007), 565-592.  doi: 10.1137/060665579.  Google Scholar

[11]

L. DupaigneM. GherguO. Goubet and G. Warnault, The Gel'fand problem for the biharmonic operator, Arch. Ration. Mech. Anal., 208 (2013), 725-752.  doi: 10.1007/s00205-013-0613-0.  Google Scholar

[12]

D. E. EdmundsD. Fortunato and E. Jannelli, Critical exponents, Critical dimensions and the biharmonic operator, Arch. Ration. Mech. Anal., 112 (1990), 269-289.  doi: 10.1007/BF00381236.  Google Scholar

[13]

R. L. 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

[14]

P. L. Felmer and A. Quaas, Fundamental solutions and two properties of elliptic maximal and minimal operators, Trans. Amer. Math. Soc., 361 (2009), 5721-5736.  doi: 10.1090/S0002-9947-09-04566-8.  Google Scholar

[15]

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

[16]

D. Gilbarg and J. Serrin, On isolated singularites of solutions of second order elliptic differential equations, J. Anal. Math., 4 (1955/56), 309-340.  doi: 10.1007/BF02787726.  Google Scholar

[17]

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

[18]

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

[19]

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

[20]

Z. M. GuoX. H. Guan and F. S. Wan, Sobolev type embedding and weak solutions with a prescribed singular set, Sci. China Math., 59 (2016), 1975-1994.  doi: 10.1007/s11425-015-0698-0.  Google Scholar

[21]

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

[22]

Z. M. GuoX. H. Guan and Y. G. Zhao, Uniqueness and asymptotic behavior of solutions of a biharmonic equation with supercritical exponent, Discrete Conti. Dyn. Syst., 39 (2019), 2613-2636.  doi: 10.3934/dcds.2019109.  Google Scholar

[23]

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

[24]

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

[25]

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

[26]

Z. M. Guo and J. C. Wei, On a fourth order elliptic problem with negative exponent, SIAM J. Math. Anal., 40 (2008/09), 2034-2054.  doi: 10.1137/070703375.  Google Scholar

[27]

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

[28]

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

[29]

D. A. Labutin, Isolated singularities for fully nonlinear elliptic equations, J. Differ. Equ., 177 (2001), 49-76.  doi: 10.1006/jdeq.2001.3998.  Google Scholar

[30]

J. Y. Li and F. S. Wan, Bôcher-type theorem on $N$-dimensioanl manifolds with conical metric, Proc. Amer. Math. Soc., 147 (2019), 4527-4538.  doi: 10.1090/proc/14554.  Google Scholar

[31]

Y. Y. Li and L. Nguyen, Harnack inequalities and Bôcher-type theorems for conformally invariant, fully nonlinear degenerate elliptic equations, Comm. Pure Appl. Math., 67 (2014), 1843-1876.  doi: 10.1002/cpa.21502.  Google Scholar

[32]

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

[33]

C. S. Lin and Z. Q. Wang, Symmetry of extremal functions for the Caffarelli-Kohn-Nirenberg inequalities, Proc. Amer. Math. Soc., 132 (2006), 1685-1691.  doi: 10.1090/S0002-9939-04-07245-4.  Google Scholar

[34]

R. Musina, Weighted Sobolev spaces of radially symmetric functions, Ann. Mat. Pura Appl., 193 (2014), 1629-1659.  doi: 10.1007/s10231-013-0348-4.  Google Scholar

[35]

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

[36]

C. Wang and D. Ye, Some Liouville theorems for Hénon type elliptic equations, J. Funct. Anal., 262 (2012), 1705-1727.  doi: 10.1016/j.jfa.2011.11.017.  Google Scholar

[37]

J. C. Wei and X. W. Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann., 313 (1999), 207-228.  doi: 10.1007/s002080050258.  Google Scholar

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