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 & 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[28]

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

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[28]

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

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