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

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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
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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.
[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.
[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.
[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.
[5]	C. Bandle and M. Marcus, The positive radial solutions of a class of semilinear elliptic equations,, J. Reine Angew. Math., 401 (1989), 25.
[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.
[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.
[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.
[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.
[10]	A. Grigor'yan, "Heat Kernel and Analysis on Manifolds'',, AMS, (2009).
[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.
[12]	P. Hartman, "Ordinary Differential Equations'',, Birkh\, (1982).
[13]	D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources,, Arch. Rational Mech. Anal., 49 (1973), 241.
[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.
[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.
[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.
[17]	G. Mancini and K. Sandeep, On a semilinear elliptic equation in $H^n$,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 7 (2008), 635.
[18]	W.-M. Ni, On the positive radial solutions of some semilinear elliptic equations on $R^n$,, Appl. Math. Optim., 9 (1983), 373.
[19]	W.-M. Ni and S. Yotsutani, Semilinear elliptic equations of Matukuma-type and related topics,, Japan J. Appl. Math., 5 (1988), 1.
[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.
[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.

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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.
[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.
[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.
[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.
[5]	C. Bandle and M. Marcus, The positive radial solutions of a class of semilinear elliptic equations,, J. Reine Angew. Math., 401 (1989), 25.
[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.
[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.
[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.
[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.
[10]	A. Grigor'yan, "Heat Kernel and Analysis on Manifolds'',, AMS, (2009).
[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.
[12]	P. Hartman, "Ordinary Differential Equations'',, Birkh\, (1982).
[13]	D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources,, Arch. Rational Mech. Anal., 49 (1973), 241.
[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.
[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.
[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.
[17]	G. Mancini and K. Sandeep, On a semilinear elliptic equation in $H^n$,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 7 (2008), 635.
[18]	W.-M. Ni, On the positive radial solutions of some semilinear elliptic equations on $R^n$,, Appl. Math. Optim., 9 (1983), 373.
[19]	W.-M. Ni and S. Yotsutani, Semilinear elliptic equations of Matukuma-type and related topics,, Japan J. Appl. Math., 5 (1988), 1.
[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.
[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.

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