American Institute of Mathematical Sciences

Advanced
March  2014, 13(2): 949-960. doi: 10.3934/cpaa.2014.13.949

Existence and uniqueness of singular solutions for elliptic equation on the hyperbolic space

Yen-Lin Wu 1, , Zhi-You Chen 2, , Jann-Long Chern 2, and  Y. Kabeya 3,

1. 

Department of Mathematics, National Central University, Chung-Li, 32001, Taiwan
2. 

Department of Mathematics, National Central University, Chung-Li 32001
3. 

Department of Mathematical Sciences, Osaka Prefecture University, 1-1 Gakuen-cho, Naka-ku, Sakai, 599-8531

Received  August 2012 Revised  June 2013 Published  October 2013

In this article, we consider the following semilinear elliptic equation on the hyperbolic space \begin{eqnarray} \Delta_{H^n} u-\lambda u+|u|^{p-1}u=0\quad on\quad H^n\setminus \{Q\} \end{eqnarray} where $\Delta_{H^n}$ is the Laplace-Beltrami operator on the hyperbolic space \begin{eqnarray} H^n=\{(x_1,\cdots, x_n,x_{n+1})|x_1^2+\cdots+x_n^2-x_{n+1}^2=-1\}, \end{eqnarray} $n>10,\ p>1, \lambda>0, $ and $Q=(0,\cdots,0,1)$. We provide the existence and uniqueness of a singular positive ``radial'' solution of the above equation for big $p$ (greater than the Joseph-Lundgren exponent, which appears if $n > 10$) as well as its asymptotic behavior.
Keywords: hyperbolic space., singular solutions, Nonlinear elliptic equations.
Mathematics Subject Classification: Primary: 35J15; Secondary: 35J61, 58J0.
Citation: Yen-Lin Wu, Zhi-You Chen, Jann-Long Chern, Y. Kabeya. Existence and uniqueness of singular solutions for elliptic equation on the hyperbolic space. Communications on Pure & Applied Analysis, 2014, 13 (2) : 949-960. doi: 10.3934/cpaa.2014.13.949
References:
[1]

S. Bae, Separation structure of positive radial solutions of a semilinear elliptic equation in $R^n$,, J. Differential Equations, 194 (2003), 460.  doi: 10.1006/jdeq.2001.4162.  Google Scholar

[2]

S. Bae and T. K. Chang, On a class of semilinear elliptic equations in $R^n$,, J. Differential Equations, 185 (2002), 225.  doi: 10.1016/s0022-396(03)00172-4.  Google Scholar

[3]

C. Bandle, A. Brillard and M. Flucher, Green's function, harmonic transplantation and best Sobolev constant in spaces of constant curvature,, Trans. Amer. Math. Soc., 350 (1998), 1103.  doi: 10.1090/S0002-9947-98-02085-6.  Google Scholar

[4]

C. Bandle and Y. Kabeya, On the positive, "radial" solutions of a semilinear elliptic equation on $H^N$,, Adv. Nonlinear Anal., 1 (2012), 1.  doi: 10.1515/ana-2011-0004.  Google Scholar

[5]

C. Bandle and M. Marcus, The positive radial solutions of a class of semilinear elliptic equations,, J. Reine Angew. Math., 401 (1989), 25.   Google Scholar

[6]

M. Bonforte, F. Gazzola, G. Grillo and J. L. Vazquez, Classification of radial solutions to the Emden-Fowler equation on the hyperbolic space,, Cal. Var. Partial Differential Equations, 46 (2013), 375.   Google Scholar

[7]

J.-L. Chern, Z.-Y. Chen, J-H. Chen and Y.-L. Tang, On the classification of standing wave solutions for the Schrödinger equation,, Comm. Partial Differential Equations, 35 (2010), 275.  doi: 10.1080/03605300903419767.  Google Scholar

[8]

B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle,, Comm. Math. Phys., 68 (1979), 209.  doi: 10.1007/BF01221125.  Google Scholar

[9]

B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^n$,, Adv. Math. Suppl. Stud., 7A (1981), 369.   Google Scholar

[10]

A. Grigor'yan, "Heat Kernel and Analysis on Manifolds'',, AMS, (2009).   Google Scholar

[11]

C. Gui, W.-M. Ni and X.-F. Wang, On the stability and instability of positive steady states of a semilinear heat equation in $R^n$,, Comm. Pure Appl. Math., 45 (1992), 1153.  doi: 10.1002/cpa.3160450906.  Google Scholar

[12]

P. Hartman, "Ordinary Differential Equations'',, Birkh\, (1982).   Google Scholar

[13]

D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources,, Arch. Rational Mech. Anal., 49 (1973), 241.   Google Scholar

[14]

S. Kumaresan and J. Prajapat, Analogue of Gidas-Ni-Nirenberg result in hyperbolic space and sphere,, Rend. Inst. Math. Univ. Trieste, 30 (1998), 107.   Google Scholar

[15]

Y. Li, Asymptotic behavior of positive solutions of equation $\Delta u + K(x)u^p = 0$ in $R^n$,, J. Differential Equations, 95 (1992), 304.  doi: 10.1016/0022-0396(92)90034-K.  Google Scholar

[16]

Y. Liu, Y. Li and Y. Deng, Separation property of solutions for a semilinear elliptic equation,, J. Differential Equations, 163 (2000), 381.  doi: 10.1006/jdeq.1999.3735.  Google Scholar

[17]

G. Mancini and K. Sandeep, On a semilinear elliptic equation in $H^n$,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 7 (2008), 635.   Google Scholar

[18]

W.-M. Ni, On the positive radial solutions of some semilinear elliptic equations on $R^n$,, Appl. Math. Optim., 9 (1983), 373.   Google Scholar

[19]

W.-M. Ni and S. Yotsutani, Semilinear elliptic equations of Matukuma-type and related topics,, Japan J. Appl. Math., 5 (1988), 1.   Google Scholar

[20]

S. Stapelkamp, The Brezis-Nirenberg problem on $H^n$: existence and uniqueness of solutions, in "Elliptic and Parabolic Problems- Rolduc and Gaeta 2001,'', Bemelmans et al. ed., (2002), 283.   Google Scholar

[21]

X.-F. Wang, On Cauchy Problem for reaction-diffusion equations,, Trans. Amer. Math. Soc., 337 (1993), 549.  doi: 10.1090/S0002-9947-1993-1153015-5.  Google Scholar

show all references

References:
[1]

S. Bae, Separation structure of positive radial solutions of a semilinear elliptic equation in $R^n$,, J. Differential Equations, 194 (2003), 460.  doi: 10.1006/jdeq.2001.4162.  Google Scholar

[2]

S. Bae and T. K. Chang, On a class of semilinear elliptic equations in $R^n$,, J. Differential Equations, 185 (2002), 225.  doi: 10.1016/s0022-396(03)00172-4.  Google Scholar

[3]

C. Bandle, A. Brillard and M. Flucher, Green's function, harmonic transplantation and best Sobolev constant in spaces of constant curvature,, Trans. Amer. Math. Soc., 350 (1998), 1103.  doi: 10.1090/S0002-9947-98-02085-6.  Google Scholar

[4]

C. Bandle and Y. Kabeya, On the positive, "radial" solutions of a semilinear elliptic equation on $H^N$,, Adv. Nonlinear Anal., 1 (2012), 1.  doi: 10.1515/ana-2011-0004.  Google Scholar

[5]

C. Bandle and M. Marcus, The positive radial solutions of a class of semilinear elliptic equations,, J. Reine Angew. Math., 401 (1989), 25.   Google Scholar

[6]

M. Bonforte, F. Gazzola, G. Grillo and J. L. Vazquez, Classification of radial solutions to the Emden-Fowler equation on the hyperbolic space,, Cal. Var. Partial Differential Equations, 46 (2013), 375.   Google Scholar

[7]

J.-L. Chern, Z.-Y. Chen, J-H. Chen and Y.-L. Tang, On the classification of standing wave solutions for the Schrödinger equation,, Comm. Partial Differential Equations, 35 (2010), 275.  doi: 10.1080/03605300903419767.  Google Scholar

[8]

B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle,, Comm. Math. Phys., 68 (1979), 209.  doi: 10.1007/BF01221125.  Google Scholar

[9]

B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^n$,, Adv. Math. Suppl. Stud., 7A (1981), 369.   Google Scholar

[10]

A. Grigor'yan, "Heat Kernel and Analysis on Manifolds'',, AMS, (2009).   Google Scholar

[11]

C. Gui, W.-M. Ni and X.-F. Wang, On the stability and instability of positive steady states of a semilinear heat equation in $R^n$,, Comm. Pure Appl. Math., 45 (1992), 1153.  doi: 10.1002/cpa.3160450906.  Google Scholar

[12]

P. Hartman, "Ordinary Differential Equations'',, Birkh\, (1982).   Google Scholar

[13]

D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources,, Arch. Rational Mech. Anal., 49 (1973), 241.   Google Scholar

[14]

S. Kumaresan and J. Prajapat, Analogue of Gidas-Ni-Nirenberg result in hyperbolic space and sphere,, Rend. Inst. Math. Univ. Trieste, 30 (1998), 107.   Google Scholar

[15]

Y. Li, Asymptotic behavior of positive solutions of equation $\Delta u + K(x)u^p = 0$ in $R^n$,, J. Differential Equations, 95 (1992), 304.  doi: 10.1016/0022-0396(92)90034-K.  Google Scholar

[16]

Y. Liu, Y. Li and Y. Deng, Separation property of solutions for a semilinear elliptic equation,, J. Differential Equations, 163 (2000), 381.  doi: 10.1006/jdeq.1999.3735.  Google Scholar

[17]

G. Mancini and K. Sandeep, On a semilinear elliptic equation in $H^n$,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 7 (2008), 635.   Google Scholar

[18]

W.-M. Ni, On the positive radial solutions of some semilinear elliptic equations on $R^n$,, Appl. Math. Optim., 9 (1983), 373.   Google Scholar

[19]

W.-M. Ni and S. Yotsutani, Semilinear elliptic equations of Matukuma-type and related topics,, Japan J. Appl. Math., 5 (1988), 1.   Google Scholar

[20]

S. Stapelkamp, The Brezis-Nirenberg problem on $H^n$: existence and uniqueness of solutions, in "Elliptic and Parabolic Problems- Rolduc and Gaeta 2001,'', Bemelmans et al. ed., (2002), 283.   Google Scholar

[21]

X.-F. Wang, On Cauchy Problem for reaction-diffusion equations,, Trans. Amer. Math. Soc., 337 (1993), 549.  doi: 10.1090/S0002-9947-1993-1153015-5.  Google Scholar

[1]

Fabio Punzo. Support properties of solutions to nonlinear parabolic equations with variable density in the hyperbolic space. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 657-670. doi: 10.3934/dcdss.2012.5.657
[2]

L. Ke. Boundary behaviors for solutions of singular elliptic equations. Conference Publications, 1998, 1998 (Special) : 388-396. doi: 10.3934/proc.1998.1998.388
[3]

Xiaomei Sun, Wenyi Chen. Positive solutions for singular elliptic equations with critical Hardy-Sobolev exponent. Communications on Pure & Applied Analysis, 2011, 10 (2) : 527-540. doi: 10.3934/cpaa.2011.10.527
[4]

J. Chen, K. Murillo, E. M. Rocha. Two nontrivial solutions of a class of elliptic equations with singular term. Conference Publications, 2011, 2011 (Special) : 272-281. doi: 10.3934/proc.2011.2011.272
[5]

Peter Poláčik, Darío A. Valdebenito. Existence of quasiperiodic solutions of elliptic equations on the entire space with a quadratic nonlinearity. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 1-25. doi: 10.3934/dcdss.2020077
[6]

Xia Huang. Stable weak solutions of weighted nonlinear elliptic equations. Communications on Pure & Applied Analysis, 2014, 13 (1) : 293-305. doi: 10.3934/cpaa.2014.13.293
[7]

Vianney Perchet, Marc Quincampoix. A differential game on Wasserstein space. Application to weak approachability with partial monitoring. Journal of Dynamics & Games, 2019, 6 (1) : 65-85. doi: 10.3934/jdg.2019005
[8]

Giuseppe Maria Coclite, Mario Michele Coclite. Positive solutions of an integro-differential equation in all space with singular nonlinear term. Discrete & Continuous Dynamical Systems - A, 2008, 22 (4) : 885-907. doi: 10.3934/dcds.2008.22.885
[9]

Françoise Demengel, O. Goubet. Existence of boundary blow up solutions for singular or degenerate fully nonlinear equations. Communications on Pure & Applied Analysis, 2013, 12 (2) : 621-645. doi: 10.3934/cpaa.2013.12.621
[10]

Xiying Sun, Qihuai Liu, Dingbian Qian, Na Zhao. Infinitely many subharmonic solutions for nonlinear equations with singular $ \phi $-Laplacian. Communications on Pure & Applied Analysis, 2020, 19 (1) : 279-292. doi: 10.3934/cpaa.20200015
[11]

Y. Efendiev, B. Popov. On homogenization of nonlinear hyperbolic equations. Communications on Pure & Applied Analysis, 2005, 4 (2) : 295-309. doi: 10.3934/cpaa.2005.4.295
[12]

Zhigang Wang, Lei Wang, Yachun Li. Renormalized entropy solutions for degenerate parabolic-hyperbolic equations with time-space dependent coefficients. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1163-1182. doi: 10.3934/cpaa.2013.12.1163
[13]

Matteo Novaga, Diego Pallara, Yannick Sire. A symmetry result for degenerate elliptic equations on the Wiener space with nonlinear boundary conditions and applications. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 815-831. doi: 10.3934/dcdss.2016030
[14]

Monica Conti, Vittorino Pata, M. Squassina. Singular limit of dissipative hyperbolic equations with memory. Conference Publications, 2005, 2005 (Special) : 200-208. doi: 10.3934/proc.2005.2005.200
[15]

Limei Dai. Entire solutions with asymptotic behavior of fully nonlinear uniformly elliptic equations. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1707-1714. doi: 10.3934/cpaa.2011.10.1707
[16]

Dagny Butler, Eunkyung Ko, Eun Kyoung Lee, R. Shivaji. Positive radial solutions for elliptic equations on exterior domains with nonlinear boundary conditions. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2713-2731. doi: 10.3934/cpaa.2014.13.2713
[17]

Y. Efendiev, Alexander Pankov. Meyers type estimates for approximate solutions of nonlinear elliptic equations and their applications. Discrete & Continuous Dynamical Systems - B, 2006, 6 (3) : 481-492. doi: 10.3934/dcdsb.2006.6.481
[18]

Sophia Th. Kyritsi, Nikolaos S. Papageorgiou. Multiple solutions for nonlinear elliptic equations with an asymmetric reaction term. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2469-2494. doi: 10.3934/dcds.2013.33.2469
[19]

Antonio Vitolo, Maria E. Amendola, Giulio Galise. On the uniqueness of blow-up solutions of fully nonlinear elliptic equations. Conference Publications, 2013, 2013 (special) : 771-780. doi: 10.3934/proc.2013.2013.771
[20]

Song Peng, Aliang Xia. Multiplicity and concentration of solutions for nonlinear fractional elliptic equations with steep potential. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1201-1217. doi: 10.3934/cpaa.2018058

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences