American Institute of Mathematical Sciences

Advanced
2013, 2013(special): 385-391. doi: 10.3934/proc.2013.2013.385

A unified approach to Matukuma type equations on the hyperbolic space or on a sphere

Yoshitsugu Kabeya 1,

1. 

Department of Mathematical Sciences, Osaka Prefecture University, Sakai, 599-8531, Japan

Received  September 2012 Revised  February 2013 Published  November 2013

In this note, we consider the following Matukuma type equation on the hyperbolic space or on a spherical cap of the unit sphere under the homogeneous Dirichlet boundary condition \begin{eqnarray*} \Lambda u+K(x)u_+^{p}=0, \end{eqnarray*} where $\Lambda$ is the Laplace-Beltrami operator on the hyperbolic space or on the unit sphere. Under suitable assumptions on $K$, we determine the structure of positive solutions.
Keywords: Matukuma type equations, spherical cap., hyperbolic space.
Mathematics Subject Classification: Primary: 35J15; Secondary: 35J61, 58J0.
Citation: Yoshitsugu Kabeya. A unified approach to Matukuma type equations on the hyperbolic space or on a sphere. Conference Publications, 2013, 2013 (special) : 385-391. doi: 10.3934/proc.2013.2013.385
References:
[1]

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.  Google Scholar

[2]

C. Bandle and Y. Kabeya, On the positive, "radial" solutions of a semilinear elliptic equation on $\mathbb H^N$, Adv. Nonlinear Anal. 1 (2012), 1-25.  Google Scholar

[3]

C. Bandle and L. A. Peletier, Best Sobolev constants and Emden equations for the critical exponent in $S^3$, Math. Ann. 313 (1999), 83-93.  Google Scholar

[4]

C. Bandle and S. Stingelin, New numerical solutions for the Brezis-Nirenberg problem on $\mathbbS^n$, "Elliptic and parabolic problems, A special Tribute to the Work of H. Brezis'', Progress in Nonlinear Differential Equations and Their Applications, 63 (2005), 13-21.  Google Scholar

[5]

C. Bandle and J. Wei, Multiple clustered layer solutions for semilinear elliptic problems on $S^n$, Commun. Partial Differential Equations 33 (2008), 613-635.  Google Scholar

[6]

C. Bandle and J. Wei, Nonradial clustered spike solutions for semilinear elliptic problems on $S^n$, J. Anal. Math. 102 (2007), 181-208.  Google Scholar

[7]

H. Brezis and L. A. Peletier, Elliptic equations with critical exponents on spherical caps of $S^3$, J. Anal. Math. 98 (2006), 279-316.  Google Scholar

[8]

M. Bonforte, F. Gazzola, G. Grillo and J. L. Vazquez, Classification of radial solutions to the Emden-Fowler equation on the hyperbolic space,, to appear., ().   Google Scholar

[9]

B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243.  Google Scholar

[10]

B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry of posiive solutions of nonlinear elliptic equations in $R^n$, Adv. Math. Suppl. Stud. 7A (1981), 369-402.  Google Scholar

[11]

A. Grigor'yan, "Heat kernel and analysis on manifolds'', AMS, Providence, (2009).  Google Scholar

[12]

Y. Kabeya, E. Yanagida and S. Yotsutani, Global structure of solutions for equations of Brezis-Nirenberg type on the unit ball, Proc. Royal Soc. Edinburgh, 131A (2001), 647-665.  Google Scholar

[13]

Y. Kabeya, E. Yanagida and S. Yotsutani, Canonical forms and structure theorems for radial solutions to semi-linear elliptic problems, Comm. Pure Appl. Anal. 1, (2002), 85-102.  Google Scholar

[14]

N. Kawanao, E. Yanagida and S. Yotsutani, Structure theorems for positive radial solutions to $\Delta u+K(|x|)u^p=0$ in $R^n$, Funkcial. Ekvac. 36 (1993), 557-579.  Google Scholar

[15]

A. Kosaka, Emden equation involving the critical Sobolev exponent with the third-kind boundary condition in $S^3$, Kodai J. Math. 35 (2012), 613-628.  Google Scholar

[16]

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.  Google Scholar

[17]

G. Mancini and K. Sandeep, On a semilinear elliptic equation in $\mathbbH^n$, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5), 7 (2008), 635-671.  Google Scholar

[18]

S. Stapelkamp, The Brezis-Nirenberg problem on $\mathbbH^n$: 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.  Google Scholar

[19]

S. Stingelin, New numerical solutions for the Brezis-Nirenberg problem on $S^n$, Universität Basel preprint 2003-15, 2003. Google Scholar

[20]

E. Yanagida and S. Yotsutani, Classifications of the structure of positive radial solutions to $\Delta u+K(|x|)u^p=0$ in $R^n$, Arch. Rational Mech. Anal. 124 (1993), 239-259.  Google Scholar

[21]

E. Yanagida and S. Yotsutani, Pohozaev identity and its applications, RIMS Kokyuroku 834 (1993), 80-90.  Google Scholar

[22]

E. Yanagida and S. Yotsutani, Existence of positive radial solutions to $\Delta u+K(|x|)u^p=0$ in $R^n$, J. Differential Equations 115, 477-502 (1995).  Google Scholar

[23]

E. Yanagida and S. Yotsutani, A unified approach to the structure of radial solutions for semi-linear elliptic problems, Japan J. Indust. Appl. Math. 18 (2001), 503-519.  Google Scholar

show all references

References:
[1]

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.  Google Scholar

[2]

C. Bandle and Y. Kabeya, On the positive, "radial" solutions of a semilinear elliptic equation on $\mathbb H^N$, Adv. Nonlinear Anal. 1 (2012), 1-25.  Google Scholar

[3]

C. Bandle and L. A. Peletier, Best Sobolev constants and Emden equations for the critical exponent in $S^3$, Math. Ann. 313 (1999), 83-93.  Google Scholar

[4]

C. Bandle and S. Stingelin, New numerical solutions for the Brezis-Nirenberg problem on $\mathbbS^n$, "Elliptic and parabolic problems, A special Tribute to the Work of H. Brezis'', Progress in Nonlinear Differential Equations and Their Applications, 63 (2005), 13-21.  Google Scholar

[5]

C. Bandle and J. Wei, Multiple clustered layer solutions for semilinear elliptic problems on $S^n$, Commun. Partial Differential Equations 33 (2008), 613-635.  Google Scholar

[6]

C. Bandle and J. Wei, Nonradial clustered spike solutions for semilinear elliptic problems on $S^n$, J. Anal. Math. 102 (2007), 181-208.  Google Scholar

[7]

H. Brezis and L. A. Peletier, Elliptic equations with critical exponents on spherical caps of $S^3$, J. Anal. Math. 98 (2006), 279-316.  Google Scholar

[8]

M. Bonforte, F. Gazzola, G. Grillo and J. L. Vazquez, Classification of radial solutions to the Emden-Fowler equation on the hyperbolic space,, to appear., ().   Google Scholar

[9]

B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243.  Google Scholar

[10]

B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry of posiive solutions of nonlinear elliptic equations in $R^n$, Adv. Math. Suppl. Stud. 7A (1981), 369-402.  Google Scholar

[11]

A. Grigor'yan, "Heat kernel and analysis on manifolds'', AMS, Providence, (2009).  Google Scholar

[12]

Y. Kabeya, E. Yanagida and S. Yotsutani, Global structure of solutions for equations of Brezis-Nirenberg type on the unit ball, Proc. Royal Soc. Edinburgh, 131A (2001), 647-665.  Google Scholar

[13]

Y. Kabeya, E. Yanagida and S. Yotsutani, Canonical forms and structure theorems for radial solutions to semi-linear elliptic problems, Comm. Pure Appl. Anal. 1, (2002), 85-102.  Google Scholar

[14]

N. Kawanao, E. Yanagida and S. Yotsutani, Structure theorems for positive radial solutions to $\Delta u+K(|x|)u^p=0$ in $R^n$, Funkcial. Ekvac. 36 (1993), 557-579.  Google Scholar

[15]

A. Kosaka, Emden equation involving the critical Sobolev exponent with the third-kind boundary condition in $S^3$, Kodai J. Math. 35 (2012), 613-628.  Google Scholar

[16]

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.  Google Scholar

[17]

G. Mancini and K. Sandeep, On a semilinear elliptic equation in $\mathbbH^n$, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5), 7 (2008), 635-671.  Google Scholar

[18]

S. Stapelkamp, The Brezis-Nirenberg problem on $\mathbbH^n$: 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.  Google Scholar

[19]

S. Stingelin, New numerical solutions for the Brezis-Nirenberg problem on $S^n$, Universität Basel preprint 2003-15, 2003. Google Scholar

[20]

E. Yanagida and S. Yotsutani, Classifications of the structure of positive radial solutions to $\Delta u+K(|x|)u^p=0$ in $R^n$, Arch. Rational Mech. Anal. 124 (1993), 239-259.  Google Scholar

[21]

E. Yanagida and S. Yotsutani, Pohozaev identity and its applications, RIMS Kokyuroku 834 (1993), 80-90.  Google Scholar

[22]

E. Yanagida and S. Yotsutani, Existence of positive radial solutions to $\Delta u+K(|x|)u^p=0$ in $R^n$, J. Differential Equations 115, 477-502 (1995).  Google Scholar

[23]

E. Yanagida and S. Yotsutani, A unified approach to the structure of radial solutions for semi-linear elliptic problems, Japan J. Indust. Appl. Math. 18 (2001), 503-519.  Google Scholar

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