July  2017, 16(4): 1189-1198. doi: 10.3934/cpaa.2017058

A critical exponent of Joseph-Lundgren type for an Hénon equation on the hyperbolic space

Mathematical Institute, Tohoku University, 6-3, Aramaki Aza Aoba, Sendai 980-8578, Japan

Received  May 2016 Revised  February 2017 Published  April 2017

We devote the present paper to studying a critical exponent with respect to the stability of solutions to Hénon type equation on the hyperbolic space. In order to specify the critical exponent, we prove the existence and nonexistence result for stable solutions. In this paper, we obtain stable, positive, and radial solutions of the Hénon type equation for the supercritical case. Moreover, we prove that the set of these stable solutions has the separation structure.

Citation: Shoichi Hasegawa. A critical exponent of Joseph-Lundgren type for an Hénon equation on the hyperbolic space. Communications on Pure and Applied Analysis, 2017, 16 (4) : 1189-1198. doi: 10.3934/cpaa.2017058
References:
[1]

A. Aftalion and F. Pacella, Qualitative properties of nodal solutions of semilinear elliptic equations in radially symmetric domains, C. R. Math. Acad. Sci. Paris, 339 (2004), 339-344.  doi: 10.1016/j.crma.2004.07.004.

[2]

S. Bae and Y. Naito, Existence and separation of positive radial solutions for semilinear elliptic equations, J. Differential Equations, 257 (2014), 2430-2463.  doi: 10.1016/j.jde.2014.05.042.

[3]

C. Bandle and Y. Kabeya, On the positive, "radial" solutions of a semilinear elliptic equation in $\mathbb{H}^N$, Adv. Nonlinear Anal., 1 (2012), 1-25.  doi: 10.1515/ana-2011-0004.

[4]

E. BerchioA. Ferrero and G. Grillo, Stability and qualitative properties of radial solutions of the Lane-Emden-Fowler equation on Riemannian models, J. Math. Pures Appl. (9), 102 (2014), 1-35.  doi: 10.1016/j.matpur.2013.10.012.

[5]

A. Bahri and P. L. Lions, Solutions of superlinear elliptic equations and their Morse indices, Comm. Pure Appl. Math., 45 (1992), 1205-1215.  doi: 10.1002/cpa.3160450908.

[6]

M. Bhakta and K. Sandeep, Poincaré-Sobolev equations in the hyperbolic space, Calc. Var. Partial Differential Equations, 44 (2012), 247-269.  doi: 10.1007/s00526-011-0433-8.

[7]

M. BonforteF. GazzolaG. Grillo and J. L. Vázquez, Classification of radial solutions to the Emden-Fowler equation on the hyperbolic space, Calc. Var. Partial Differential Equations, 46 (2013), 375-401.  doi: 10.1007/s00526-011-0486-8.

[8]

E. N. DancerY. Du and Z. Guo, Finite Morse index solutions of an elliptic equation with supercritical exponent, J. Differential Equations, 250 (2011), 3281-3310.  doi: 10.1016/j.jde.2011.02.005.

[9]

Y. Du and Z. Guo, Finite Morse index solutions of weighted elliptic equations and the critical exponents, Calc. Var. Partial Differential Equations, 54 (2015), 3161-3181.  doi: 10.1007/s00526-015-0897-z.

[10]

A. Farina, On the classification of solutions of the Lane-Emden equation on unbounded domains of $\mathbb{R}^N$, J. Math. Pures Appl. (9), 87 (2007), 537-561.  doi: 10.1016/j.matpur.2007.03.001.

[11]

B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations, 6 (1981), 883-901.  doi: 10.1080/03605308108820196.

[12]

B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34 (1981), 525-598.  doi: 10.1002/cpa.3160340406.

[13]

C. GuiW.-M. Ni and X. Wang, On the stability and instability of positive steady states of a semilinear heat equation in $\mathbb{R}^N$, Comm. Pure Appl. Math., 45 (1992), 1153-1181.  doi: 10.1002/cpa.3160450906.

[14]

C. GuiW.-M. Ni and X. Wang, Further study on a nonlinear heat equation, J. Differential Equations, 169 (2001), 588-613.  doi: 10.1006/jdeq.2000.3909.

[15]

S. Hasegawa, A critical exponent for Hénon type equation on the hyperbolic space, Nonlinear Anal., 129 (2015), 343-370.  doi: 10.1016/j.na.2015.09.013.

[16]

H. He, The existence of solutions for Hénon equation in hyperbolic space, Proc. Japan Acad. Ser. A Math. Sci., 89 (2013), 24-28.  doi: 10.3792/pjaa.89.24.

[17]

D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal., 49 (1972/73), 241-269.  doi: 10.1007/BF00250508.

[18]

Y. LiuY. 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.

[19]

G. Mancini and K. Sandeep, On a semilinear elliptic equation in Hn, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 7 (2008), 635-671. 

[20]

W.-M. Ni and S. Yotsutani, Semilinear elliptic equations of Matukuma-type and related topics, Japan J. Appl. Math., 5 (1988), 1-32.  doi: 10.1007/BF03167899.

[21]

F. Pacella and T. Weth, Symmetry of solutions to semilinear elliptic equations via Morse index, Proc. Amer. Math. Soc., 135 (2007), 1753-1762.  doi: 10.1090/S0002-9939-07-08652-2.

[22]

P. Poláčik and E. Yanagida, On bounded and unbounded global solutions of a supercritical semilinear heat equation, Math. Ann., 327 (2003), 745-771.  doi: 10.1007/s00208-003-0469-y.

[23]

P. Poláčik and E. Yanagida, A Liouville property and quasiconvergence for a semilinear heat equation, J. Differential Equations, 208 (2005), 194-214.  doi: 10.1016/j.jde.2003.10.019.

[24]

F. Punzo, On well-posedness of semilinear parabolic and elliptic problems in the hyperbolic space, J. Differential Equations, 251 (2011), 1972-1989.  doi: 10.1016/j.jde.2011.05.033.

[25]

S. Stapelkamp, The Brézis-Nirenberg problem on $\mathbb{H}^n$: Existence and uniqueness of solutions, in Elliptic and Parabolic Problems (eds. B. Josef and Author 7), World Sci. Publ. , River Edge, NJ, (2002), 283-290. doi: 10.1142/9789812777201_0027.

[26]

S. Tanaka, Morse index and symmetry-breaking for positive solutions of one-dimensional Hénon type equations, J. Differential Equations, 255 (2013), 1709-1733.  doi: 10.1016/j.jde.2013.05.029.

[27]

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.

[28]

X. Wang, On the Cauchy problem for reaction-diffusion equations, Trans. Amer. Math. Soc., 337 (1993), 549-590.  doi: 10.2307/2154232.

show all references

References:
[1]

A. Aftalion and F. Pacella, Qualitative properties of nodal solutions of semilinear elliptic equations in radially symmetric domains, C. R. Math. Acad. Sci. Paris, 339 (2004), 339-344.  doi: 10.1016/j.crma.2004.07.004.

[2]

S. Bae and Y. Naito, Existence and separation of positive radial solutions for semilinear elliptic equations, J. Differential Equations, 257 (2014), 2430-2463.  doi: 10.1016/j.jde.2014.05.042.

[3]

C. Bandle and Y. Kabeya, On the positive, "radial" solutions of a semilinear elliptic equation in $\mathbb{H}^N$, Adv. Nonlinear Anal., 1 (2012), 1-25.  doi: 10.1515/ana-2011-0004.

[4]

E. BerchioA. Ferrero and G. Grillo, Stability and qualitative properties of radial solutions of the Lane-Emden-Fowler equation on Riemannian models, J. Math. Pures Appl. (9), 102 (2014), 1-35.  doi: 10.1016/j.matpur.2013.10.012.

[5]

A. Bahri and P. L. Lions, Solutions of superlinear elliptic equations and their Morse indices, Comm. Pure Appl. Math., 45 (1992), 1205-1215.  doi: 10.1002/cpa.3160450908.

[6]

M. Bhakta and K. Sandeep, Poincaré-Sobolev equations in the hyperbolic space, Calc. Var. Partial Differential Equations, 44 (2012), 247-269.  doi: 10.1007/s00526-011-0433-8.

[7]

M. BonforteF. GazzolaG. Grillo and J. L. Vázquez, Classification of radial solutions to the Emden-Fowler equation on the hyperbolic space, Calc. Var. Partial Differential Equations, 46 (2013), 375-401.  doi: 10.1007/s00526-011-0486-8.

[8]

E. N. DancerY. Du and Z. Guo, Finite Morse index solutions of an elliptic equation with supercritical exponent, J. Differential Equations, 250 (2011), 3281-3310.  doi: 10.1016/j.jde.2011.02.005.

[9]

Y. Du and Z. Guo, Finite Morse index solutions of weighted elliptic equations and the critical exponents, Calc. Var. Partial Differential Equations, 54 (2015), 3161-3181.  doi: 10.1007/s00526-015-0897-z.

[10]

A. Farina, On the classification of solutions of the Lane-Emden equation on unbounded domains of $\mathbb{R}^N$, J. Math. Pures Appl. (9), 87 (2007), 537-561.  doi: 10.1016/j.matpur.2007.03.001.

[11]

B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations, 6 (1981), 883-901.  doi: 10.1080/03605308108820196.

[12]

B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34 (1981), 525-598.  doi: 10.1002/cpa.3160340406.

[13]

C. GuiW.-M. Ni and X. Wang, On the stability and instability of positive steady states of a semilinear heat equation in $\mathbb{R}^N$, Comm. Pure Appl. Math., 45 (1992), 1153-1181.  doi: 10.1002/cpa.3160450906.

[14]

C. GuiW.-M. Ni and X. Wang, Further study on a nonlinear heat equation, J. Differential Equations, 169 (2001), 588-613.  doi: 10.1006/jdeq.2000.3909.

[15]

S. Hasegawa, A critical exponent for Hénon type equation on the hyperbolic space, Nonlinear Anal., 129 (2015), 343-370.  doi: 10.1016/j.na.2015.09.013.

[16]

H. He, The existence of solutions for Hénon equation in hyperbolic space, Proc. Japan Acad. Ser. A Math. Sci., 89 (2013), 24-28.  doi: 10.3792/pjaa.89.24.

[17]

D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal., 49 (1972/73), 241-269.  doi: 10.1007/BF00250508.

[18]

Y. LiuY. 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.

[19]

G. Mancini and K. Sandeep, On a semilinear elliptic equation in Hn, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 7 (2008), 635-671. 

[20]

W.-M. Ni and S. Yotsutani, Semilinear elliptic equations of Matukuma-type and related topics, Japan J. Appl. Math., 5 (1988), 1-32.  doi: 10.1007/BF03167899.

[21]

F. Pacella and T. Weth, Symmetry of solutions to semilinear elliptic equations via Morse index, Proc. Amer. Math. Soc., 135 (2007), 1753-1762.  doi: 10.1090/S0002-9939-07-08652-2.

[22]

P. Poláčik and E. Yanagida, On bounded and unbounded global solutions of a supercritical semilinear heat equation, Math. Ann., 327 (2003), 745-771.  doi: 10.1007/s00208-003-0469-y.

[23]

P. Poláčik and E. Yanagida, A Liouville property and quasiconvergence for a semilinear heat equation, J. Differential Equations, 208 (2005), 194-214.  doi: 10.1016/j.jde.2003.10.019.

[24]

F. Punzo, On well-posedness of semilinear parabolic and elliptic problems in the hyperbolic space, J. Differential Equations, 251 (2011), 1972-1989.  doi: 10.1016/j.jde.2011.05.033.

[25]

S. Stapelkamp, The Brézis-Nirenberg problem on $\mathbb{H}^n$: Existence and uniqueness of solutions, in Elliptic and Parabolic Problems (eds. B. Josef and Author 7), World Sci. Publ. , River Edge, NJ, (2002), 283-290. doi: 10.1142/9789812777201_0027.

[26]

S. Tanaka, Morse index and symmetry-breaking for positive solutions of one-dimensional Hénon type equations, J. Differential Equations, 255 (2013), 1709-1733.  doi: 10.1016/j.jde.2013.05.029.

[27]

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.

[28]

X. Wang, On the Cauchy problem for reaction-diffusion equations, Trans. Amer. Math. Soc., 337 (1993), 549-590.  doi: 10.2307/2154232.

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