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

Yen-Lin Wu; Zhi-You Chen; Jann-Long Chern; Y. Kabeya

doi:10.3934/cpaa.2014.13.949

Article Contents

2014, Volume 13, Issue 2: 949-960. Doi: 10.3934/cpaa.2014.13.949

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

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

1.
Department of Mathematics, National Central University, Chung-Li, 32001
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: July 31, 2012

Revised: May 31, 2013

Published: February 2014

Abstract Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: Primary: 35J15; Secondary: 35J61, 58J05.

Citation:

Related Papers

Cited by

References

[1]	S. Bae, Separation structure of positive radial solutions of a semilinear elliptic equation in $R^n$, J. Differential Equations, 194 (2003), 460-499.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), 225-250.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), 1103-1128.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), 1-25.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), 25-59.
[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), 375-401.
[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), 275-301;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), 209-243.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), 369-402.
[10]	A. Grigor'yan, "Heat Kernel and Analysis on Manifolds'', AMS, Providence, 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), 1153-1181.doi: 10.1002/cpa.3160450906.
[12]	P. Hartman, "Ordinary Differential Equations'', Birkhäuser, Boston, second edition, 1982.
[13]	D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal., 49 (1973), 241-269.
[14]	S. Kumaresan and J. Prajapat, Analogue of Gidas-Ni-Nirenberg result in hyperbolic space and sphere, Rend. Inst. Math. Univ. Trieste, 30 (1998), 107-112.
[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), 304-330.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), 381-406.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), 635-671.
[18]	W.-M. Ni, On the positive radial solutions of some semilinear elliptic equations on $R^n$, Appl. Math. Optim., 9 (1983), 373-380.
[19]	W.-M. Ni and S. Yotsutani, Semilinear elliptic equations of Matukuma-type and related topics, Japan J. Appl. Math., 5 (1988), 1-32.
[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., World Scientific Publ. River Edge, NJ, (2002), 283-290.
[21]	X.-F. Wang, On Cauchy Problem for reaction-diffusion equations, Trans. Amer. Math. Soc., 337 (1993), 549-590.doi: 10.1090/S0002-9947-1993-1153015-5.