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

Previous Article
Quasilinear retarded differential equations with functional dependence on piecewise constant argument
CPAA Home
This Issue
Next Article
Periodic solutions of the Brillouin electron beam focusing equation

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

Abstract
Related Papers

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

American Institute of Mathematical Sciences